Sensitivity analysis of inefficient units in data envelopment analysis

Mathematical and Computer Modelling 53 (2011) 587–596

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Sensitivity analysis of inefficient units in data envelopment analysis G.R. Jahanshahloo a , F. Hosseinzadeh Lotfi a , N. Shoja b , A. Gholam Abri b,∗ , M. Fallah Jelodar b , Kamran Jamali Firouzabadi b a

Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran

b

Department of Mathematics, Firoozkoh branch, Islamic Azad University, Firoozkoh, Iran

article

info

Article history: Received 23 June 2009 Received in revised form 19 September 2010 Accepted 20 September 2010 Keywords: Data envelopment analysis (DEA) Sensitivity Efficiency Necessary Change Region

abstract One important issue in DEA which has been studied by many DEA researchers is the sensitivity of the results of an analysis to perturbations in the data. This paper develops a procedure for performing a sensitivity analysis of the inefficient decision making units (DMUs). The procedure yields an exact ‘‘Necessary Change Region’’ in which the efficiency score of a specific inefficient DMU changes to a defined efficiency score. In what follows, we identify a new frontier, and prove the efficiency score of each arbitrary unit on it which is defined as the efficiency score. © 2011 Published by Elsevier Ltd

1. Introduction Data envelopment analysis (DEA) introduced by Charnes et al. [1] (CCR) and extended by Banker et al. [2] (BCC), is a useful method to evaluate the relative efficiency of multiple-input and multiple-output units based on the data observed. The sensitivity analysis has received great attention from researchers in recent years and so much research has been carried out in this regard. Sensitivity analysis in DEA has been deliberated from various points of view. One important issue in DEA which has been studied by many DEA researchers, is the sensitivity analysis of a specific decision making unit (DMU) which is under evaluation [3–7]. Another type of DEA sensitivity analysis is based on the superefficiency DEA approach in which the DMU under evaluation is not included in the reference set [8–12]. Charnes et al. developed a super-efficiency DEA sensitivity analysis technique for the situation where simultaneous proportional change is assumed in all inputs and outputs for a specific DMU under consideration [13,14]. DEA sensitivity analysis methods that we have just reviewed are all developed for the situation where data variations are only applied to the efficient DMU under evaluation and the data for the remaining DMUs are assumed fixed. While the sensitivity analysis of an efficient unit’s classification has been extensively studied, the issue of an inefficient unit’s estimation and classification seems to be ignored. This paper focusses on inefficient sensitivity analysis DMUs. So, the aim is to research ways to improve the inefficient units using another strategy, in addition to the evaluation of DMUs and classifying them into efficient and inefficient. The improvement is usually possible, but sometimes, reaching to the efficiency frontier and achieving the score 1 in efficiency by inefficient units are really impossible. Our objective is to reach to the efficiency score of those inefficient units whose efficiency score is less than a fixed constant α to α . (This constant is usually close to 1 and is defined by the manager). It means that if we suppose the efficiency score of inefficient unit to be θo∗ and θo∗ < α < 1, then after these variations, it will meet the efficiency score of α , and an improvement of α − θo∗ in efficiency is obtained.

∗

Corresponding author. Tel.: +98 912 3043297. E-mail address: [email protected] (A. Gholam Abri).

0895-7177/$ – see front matter © 2011 Published by Elsevier Ltd doi:10.1016/j.mcm.2010.09.008

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The variations region of every inefficient unit is called ‘‘Necessary Change Region’’. In what follows, some new frontiers are defined and with the help of some theorems, we will prove that the efficiency score of each unit of the new frontier is α . In fact, as the efficiency score of all points on the main frontier is supposed to be 1, the efficiency score on the new frontiers is α . This paper proceeds as follows. Section 2 discusses the basic DEA models. Section 3 develops a proposed method for finding the ‘‘Necessary Change Region’’. Section 4 provides a numerical example and finally, conclusions are given in Section 5. 2. Background Data Envelopment Analysis (DEA) is a technique that has been used widely in the literature of the supply chain management. This non-parametric, multi-factor approach enhances our ability to capture the multi-dimensionality of performance discussed earlier. More formally, DEA is a mathematical programming technique for measuring the relative efficiency of decision making units (DMUs) where each DMU has a set of inputs used to produce a set of outputs. Consider DMUj , (j = 1, . . . , n), where each DMU consumes m inputs to produce s outputs. Suppose that the observed input and output vectors of DMUj are Xj = (x1j , . . . , xmj ) and Yj = (y1j , . . . , ysj ) respectively, and let Xj ≥ 0 and Xj ̸= 0 and Yj ≥ 0 and Yj ̸= 0. The production possibility set Tc is defined by:

 Tc =

(X , Y ) | X ≥

n −

λ j Xj , Y ≤

j=1

n −

 λj Yj , λj ≥ 0, j = 1, . . . , n .

j =1

The above definition implies that the CCR model is as follows: min s.t.

θ n −

λj xij ≤ θ xio ,

i = 1, . . . , m

j=1 n

−

(1)

λj yrj ≥ yro ,

j=1

λj ≥ 0,

r = 1, . . . , s

j = 1, . . . , n.

Moreover, the production possibility set Tv is defined by:

 Tv =

(X , Y ) | X ≥

n −

λ j Xj , Y ≤

j=1

n − j =1

λj Yj ,

n −

 λj = 1, λj ≥ 0, j = 1, . . . , n .

j =1

The above definition implies that the BCC model is as follows: min s.t.

θ n −

λj xij ≤ θ xio ,

i = 1, . . . , m

j=1 n

−

λj yrj ≥ yro ,

r = 1, . . . , s

(2)

j=1 n

−

λj = 1,

j =1

λj ≥ 0,

j = 1, . . . , n.

In addition, the multiplier forms of the CCR, BCC models are: CCR model max

s −

ur yro

r =1 s

s.t.

−

ur yrj −

r =1 m

− i =1

m −

vi xij ≤ 0,

i=1

vi xio = 1

vi ≥ 0, ur ≥ 0 ,

i = 1, . . . , m r = 1 , . . . , s.

j = 1, . . . , n (3)

G.R. Jahanshahloo et al. / Mathematical and Computer Modelling 53 (2011) 587–596

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BCC model s −

max

ur yro + uo

r =1 s

−

s.t.

ur yrj −

m −

r =1 m

−

vi xij + uo ≤ 0,

j = 1, . . . , n (4)

i=1

vi xio = 1

i=1

vi ≥ 0 , ur ≥ 0,

i = 1, . . . , m r = 1, . . . , s.

We know that, in most models of DEA, the efficiency score of the best performers is one. To discriminate between these efficient DMUs, many methods have been suggested. One of the most important models for ranking extreme efficient units was proposed by Andersen and Petersen (AP), [15]. This model is:

θo n −

AP : min s.t.

λj xij ≤ θo xio ,

i = 1, . . . , m

j=1,j̸=o n

−

λj yrj ≥ yro ,

r = 1, . . . , s

(5)

j=1,j̸=o n

−

λj = 1

j=1,j̸=o

λj ≥ 0,

j = 1, . . . , n, j ̸= o.

Definition 1 (Reference Set). For a DMUo , we define its reference set Eo to be: Eo = {j | λ∗j > 0} in some optimal solution to (1) or (2) [16]. Definition 2 (Pareto–Koopmans Efficiency). A DMU is fully efficient, if and only if it is not possible to improve any input or output without worsening some other input or output [16]. Definition 3. A DMUo is extreme efficient, if and only if it satisfies the following two conditions: (i) It is efficient (Pareto–Koopmans Efficient). (ii) | Eo |= 1. Definition 4. A DMUo is non-extreme efficient, if and only if it satisfies the following two conditions: (i) It is efficient (Pareto–Koopmans Efficient). (ii) | Eo |> 1 (that is the CCR envelopment model corresponding DMUo has alternate optimal). 3. Proposed model In this method, we suppose that the DMUo which is under DMU evaluation, is inefficient. Moreover, we suppose the efficiency score of DMUo is less than 1 and (θo∗ < α < 1). In addition, assume α is a constant number which meets the requirement of the following circumstances. (α is a constant number and probably close to 1 defined by the manager.) What we wish, is to define a region for DMUo whose efficiency score becomes α after these changes. It means that the efficiency score of DMUo has an improvement for α − θo∗ . This region of changes of any inefficient unit whose efficiency score is smaller than α , is called the ‘‘Necessary Change Region’’. For this purpose, we first apply model (AP) for DMUj (j = 1, . . . , n) and we find all extreme efficient DMUs. In this case either efficiency scores greater than 1 are obtained for these DMUs, or (AP) is infeasible [17]. Let the set of extreme efficient DMUs in Tv be E. Having determined E, we define E ′ as follows: ′



E =

(Xj , Yj ) | (Xj , Yj ) = ′

′

′

′



1

α

Xj , Yj





,j ∈ E .

We introduce the new production possibility set Tv′ :

 Tv′

= (X , Y ) | X ≥ ′

′

′

1−

α

j∈E

 λj Xj , Y ≤ ′

− j∈E

λj Yj ,

− j∈E

λj = 1, λj ≥ 0, j ∈ E .

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G.R. Jahanshahloo et al. / Mathematical and Computer Modelling 53 (2011) 587–596

Fig. 1. Different strategies.

In the sequel, applying the BCC multiplier model for the members of E ′ , the supporting hyperplanes of the Tv′ are found. For more details and the method of finding all supporting hyperplanes of production possibility set, see [18]. On the other hand, we know that DMUo is inefficient and θo∗ < α < 1. So we are going to move toward the Tv′ frontier. Choosing different strategies (Input Oriented, Output Oriented, Combination Oriented, . . . ), the DMUo can be moved toward the frontier in different ways. To illustrate the subject, consider the following example. In Fig. 1, consider Tv , Tv′ and inefficient DMUo (θo∗ < α < 1). What we want, is the change region to bring DMUo to the Tv′ frontier. So, the efficiency score of DMUo would be improved for α − θo∗ . Obviously, as depicted in Fig. 1, by choosing different strategies, it is possible to arrive DMUo to the new frontiers. Below, by some theorems, we show that every point on Tv′ frontier has an efficiency score of α . Theorem 1. The efficiency score of each point of E ′ in Tv is α . Proof. Let M ′ with coordinates (XM ′ , YM ′ ) be an arbitrary point in E ′ . Evaluating M ′ with the BCC model in Tv , we obtain: min s.t.

θM ′ − j∈E − j∈E −

λj Xj ≤ θM ′ XM ′ = θM ′



1

α

 XM

,

λj Yj ≥ YM ′ = YM , λj = 1,

j∈E

λj ≥ 0,

j ∈ E.

This model has a feasible solution (θM ′ = α, λM = 1, λj = 0, (j ∈ E , j ̸= M )). ∗ ∗ Hence the optimal θM ′ , denoted θM ′ , is not greater than α. (θM ′ ≤ α). ∗ In this way, it will be represented that: (θM ′ ̸< α). ∗ In contradiction, assume that (θM ′ < α). ∗ So, θM ′ = α − ϵ for some ϵ > 0. From the model (6), we get:

    −  α−ϵ ϵ 1 ∗ ∗  ′ λj Xj ≤ θM ′ XM = θM ′ XM = XM = 1 − XM    α α α  j∈E  −   λY ≥Y ′ =Y j j

M

j∈E −    λj = 1      j∈E λj ≥ 0, j ∈ E .

M

(6)

G.R. Jahanshahloo et al. / Mathematical and Computer Modelling 53 (2011) 587–596

591

We know that (1 − αϵ ) < 1. So a feasible solution for DMUM is: (θM = (1 − αϵ ) < 1, λM = 1, λj = 0, (j ∈ E , j ̸= M )). Therefore, DMUM is inefficient and this is in contradiction with the assumption. ∗ Hence we have: θM  ′ = α and this completes the proof. Theorem 2. M ′ ∈ E ′ if and only if M ′ is an extreme efficient unit in Tv′ . Proof. Let M ′ with with coordinates (XM ′ , YM ′ ) be an arbitrary extreme efficient point in E ′ . As far as we know, (XM , YM ) ∈ E. At the first step, we are going to prove M ′ is efficient in Tv′ . By contradiction, let M ′ not be efficient in Tv′ . Then, we evaluate M ′ by the BCC model in Tv′ , as follows: min s.t.

θM ′ −

λj Xj′ ≤ θM ′ XM ′ = θM ′

j∈E ′

−



1

α

 XM

,

λj Yj′ ≥ YM ′ = YM ,

(7)

j∈E ′

−

λj = 1,

j∈E ′

λj ≥ 0 ,

j ∈ E′.

∗ ∗ Suppose that the optimal solution of above mentioned problem to be (λ∗ , θM ′ ). By contradiction, we assume θM ′ < 1, so the conclusion would be:

− ∗ ′ λj Xj ≤ θM∗ ′ XM ′    ′  j ∈ E  −    λ∗j Yj′ ≥ YM ′ ′ j∈E −    λ∗j = 1    ′ j ∈ E   ∗ λj ≥ 0 , j ∈ E ′ . By multiplying the first constraint to α , it results in:

− ∗ λj (α Xj′ ) ≤ θM∗ ′ (α XM ′ )    ′  j ∈ E  −    λ∗j Yj′ ≥ YM ′ j∈E ′ −    λ∗j = 1    ′   j∈∗E λj ≥ 0 , j ∈ E ′ . According to definition E and E ′ :

− ∗ λ Xj ≤ θM∗ ′ XM    j∈E j   −    λ∗j Yj ≥ YM j∈E −    λ∗j = 1      j∈∗E λj ≥ 0, j ∈ E . ∗ So, the last model has a feasible solution (λ∗ , θM ′ < 1) for the corresponding problem of M ∈ E. Hence the optimal θM , ∗ denoted by θM , is less than 1 and this is in contradiction with M ∈ E. Therefore, M ′ is efficient in Tv′ and this completes the first step of the proof.

In continue, we are going to represent that M ′ is also an extreme unit in Tv′ . In contradiction, suppose M ′ to be a non-extreme efficient unit in Tv′ . Let the set of extreme points in Tv′ be: {(X1′ , Y1′ ), . . . , (Xt′ , Yt′ )}. It would be concluded:

(XM ′ , YM ′ ) =

t − j=1

λj (Xj′ , Yj′ ),

t − j =1

λj = 1,

λj ≥ 0, j = 1, . . . , t

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G.R. Jahanshahloo et al. / Mathematical and Computer Modelling 53 (2011) 587–596

or



1

α

XM , YM

t −

 =



λj

1

α

j=1

Xj , Yj



t −

,

λj = 1,

λj ≥ 0, j = 1, . . . , t .

j=1

That is:



1

α

XM , YM



 =

t 1−

α

t −

λj Xj ,

j =1



t −

λ j Xj ,

j =1

λj = 1,

λj ≥ 0, j = 1, . . . , t .

j =1

It can be concluded from the recent equation: XM =

t −

λj Xj ,

YM =

t −

j =1

t −

λj Xj ,

j =1

λj = 1,

λj ≥ 0, j = 1, . . . , t .

j=1

Therefore:

 (XM , YM ) =

t −

λj Xj ,

j =1

t −

 λj Yj

=

t −

j =1

t −

λj (Xj , Yj ),

j =1

λj = 1,

λj ≥ 0, j = 1, . . . , t .

j =1

Thus, we have represented M with (XM , YM ) as a convex combination of extreme efficient points in Tv and this is in contradiction with our assumption. Hence, M ′ ∈ E ′ is an extreme efficient unit in Tv′ . Conversely, Let M ′ with coordinates (XM ′ , YM ′ ) be an arbitrary extreme efficient point in Tv′ . We first claim: (α XM ′ , YM ′ ) is an extreme efficient point in Tv . Assume conversely that (α XM ′ , YM ′ ) is not in E. So,

(α XM ′ , YM ′ ) =

−

−

λj (Xj , Yj ),

λj = 1,

λj ≥ 0, j ∈ E

j∈E

j∈E

or:



 −

(α XM ′ , YM ′ ) =

λj Xj ,

−

−

λj Yj ,

j∈E

j∈E

λj = 1,

λj ≥ 0, j ∈ E .

j∈E

It can be concluded:

α XM ′ =

−

λj Xj ,

YM ′ =

−

−

λj Yj ,

λj = 1,

λ j ≥ 0, j ∈ E .

j∈E

j∈E

j∈E

Therefore, we obtain:

=

XM ′

−

λj



1

α

j∈E

 Xj

,

YM ′ =

−

λj Yj ,

j∈E

−

λj = 1,

λj ≥ 0, j ∈ E .

j∈E

On the other hand, we have:

 X ′ = 1 X j j α Y ′ = Y . j

j

So, XM ′ =

−

λj Xj′ ,

YM ′ =

j∈E ′

−

−

λj Yj′ ,

j∈E ′

λj = 1,

λj ≥ 0, j ∈ E ′

j∈E ′

or:

 (XM ′ , YM ′ ) =

 − j∈E ′

λj Xj , ′

− j∈E ′

λj Yj

′

=

− j∈E ′

λj (Xj′ , Yj′ ),

−

λj = 1,

λj ≥ 0, j ∈ E ′ .

j∈E ′

Thus, we have represented (XM ′ , YM ′ ) as a convex combination of extreme efficient points in Tv′ and this is in contradiction with our assumption. Hence (α XM ′ , YM ′ ) ∈ E. Consequently, by definition of E ′ we will have: ( α1 (α XM ′ ), YM ′ ) ∈ E ′ . So, M ′ with coordinates (XM ′ , YM ′ ) ∈ E ′ and this completes the proof. 

G.R. Jahanshahloo et al. / Mathematical and Computer Modelling 53 (2011) 587–596

593

Lemma 1. If M ′ is an arbitrary non-extreme efficient point on the Tv′ frontier,then there will be a point on the Tv frontier like M, as we have: (XM ′ , YM ′ ) = ( α1 XM , YM ). Attention 1. According to Theorem 2, there is a one-to-one correspondence between E and E ′ . So M is a non-extreme efficient unit on the Tv frontier in Lemma 1. Proof. Respecting to the subject that M ′ is an arbitrary non-extreme efficient point on the Tv′ frontier, there are at least two extreme efficient points like A′ and B′ in Tv′ so that:



XM ′ = λXA′ + (1 − λ)XB′ YM ′ = λYA′ + (1 − λ)YB′ .

On the other hand, we have:

 

X A′ =

1

α

XA

 ′ YA = YA  1  ′ XB = XB α  ′ YB = YB . Therefore:



YM ′

1

(λXA + (1 − λ)XB ) α = λYA + (1 − λ)YB .

XM ′ =

The point M with coordinates (XM = λXA + (1 − λ)XB , YM = λYA + (1 − λ)YB ) will be considered. We are going to represent M to be on the Tv frontier. By contradiction, we suppose that it is not so, then, we evaluate M by the BCC model in Tv as follows: min s.t.

θM − j∈E − j∈E −

λj Xj ≤ θM XM = θM (λXA + (1 − λ)XB ) λj Yj ≥ YM = λYA + (1 − λ)YB

(8)

λj = 1 ,

j∈E

λj ≥ 0 ,

j ∈ E.

Assume that (λ , θM ) is the optimal solution for the previous problem. Respecting to contradiction assumption, it can be ∗ obviously seen that θM < 1, so: ∗

∗

− ∗ λj Xj ≤ θM∗ (λXA + (1 − λ)XB )     j ∈ E  −    λ∗ Y ≥ λY + (1 − λ)Y j

j

A

B

j∈E −    λ∗j = 1    j∈E   ∗ λj ≥ 0, j ∈ E .

By multiplying the first constraint to α1 , that result is:

  −  1  1 ∗ ∗  X ≤ θ (λ X + ( 1 − λ) X ) λ  j A B M j   α α  j∈E  −  ∗  λj Yj ≥ λYA + (1 − λ)YB j∈E −    λ∗j = 1      j∈∗E λj ≥ 0 , j ∈ E or

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− ∗ ′ λ X ≤ θM∗ XM ′    j∈E ′ j j   −    λ∗ Y ≥ Y ′ j

j

M

j∈E ′

−    λ∗j = 1    ′   j∈∗E λj ≥ 0, j ∈ E ′ . ∗ Therefore, the last model has a feasible solution (λ∗ , θM < 1) for the corresponding problem of M ′ ∈ Tv′ and this is in contradiction with M ′ that is a non-extreme efficient unit on the Tv′ frontier. Hence, our proof will be completed. 

Attention 2. Theorem 2 and Lemma 1 prove that there is a one-to-one correspondence between Tv and Tv′ frontier points. Theorem 3. The efficiency score of each point on the Tv′ frontier is α in Tv . Proof. Let M ′ be an arbitrary point on the Tv′ frontier. Each arbitrary point on the Tv′ frontier is extreme efficient or a nonextreme efficient one. (1) If M ′ is extreme efficient in Tv′ , Theorem 2 shows that M ′ ∈ E ′ . Since the efficiency score of each point of E ′ is α in Tv by Theorem 1, we have θM ′ = α in Tv . (2) If DMUM ′ is an arbitrary non-extreme efficient point on the Tv′ frontier, so there is a point on the Tv frontier according to Lemma 1 which:

 

XM ′ =



YM ′ = YM .

1

α

XM

Now, we evaluate M ′ by the BCC model in Tv as follows: min s.t.

θ− M′ j∈E −

λj Xj ≤ θM ′ XM ′ λj Yj ≥ YM ′

(9)

j∈E

−

λj = 1,

j∈E

λj ≥ 0,

j ∈ E.

∗ ∗ ∗ Let an optimal solution for last model be (λ∗ , θM ′ ). We have to prove that θM ′ = α . By contradiction, suppose that θM ′ < α . ∗ So, θM ′ = α − ϵ for some ϵ > 0. From the above model we will get:

   − ϵ α−ϵ ∗ ∗  ′ = X = 1 − λ XM X ≤ θ X ′  M j M j M   α α  j∈E  −   λ∗ Y ≥ Y ′ = Y j

j

M

M

j∈E −    λ∗j = 1     j ∈ E  ∗ λj ≥ 0, j ∈ E .

But we know that (1 − αϵ ) < 1. So, a feasible solution for M is:

   ϵ θM = 1 − < 1, λM = 1, λj = 0, (j ∈ E , j ̸= M ) . α Hence, M is inefficient and this is in contradiction with the assumption. ∗ So, θM  ′ = α in Tv and our proof will be completed. 4. A numerical example Consider seven DMUs with a single input and output. The data and results are summarized in Table 1 and are shown in Fig. 2: By Fig. 2 and using the BCC model, the units A, B, C , D and F are efficient in the BCC model. Moreover, M and N are inefficient.

G.R. Jahanshahloo et al. / Mathematical and Computer Modelling 53 (2011) 587–596

595

Table 1 Data and results. DMUs

A

B

C

D

F

M

N

Input Output Results

1 1 1.0000

2 3 1.0000

3 5 1.0000

4 6 1.0000

6 7 1.0000

3 2 0.5000

4.5 5.5 0.7778

Fig. 2. Data set in Tv and Tv′ .

Assume α = 0.800. Using the (AP) model, efficiency scores which are greater than 1 are obtained for the units A, C , D and F . Hence, these DMUs are extreme efficient and E = {A(1, 1), C (3, 5), D(4, 6), F (6, 7)}.  Then, by definition of E ′ we will obtain E ′ = A′ (1.25, 1), C ′ (3.75, 5), D′ (5, 6), F ′ (7.5, 7) . Now, applying the BCC multiplier model for the members of E ′ , we find the supporting hyperplanes of the Tv′ . These hyperplanes are: H1 = {(x, y)|5y − 8x = 5} H2 = {(x, y)|10y − 8x = 20} H3 = {(x, y)|5y − 2x = 20}. On the other hand, we know that DMUM and DMUN are inefficient and

θM∗ = 0.50000 < α = 0.800,

θN∗ = 0.7778 < α = 0.800.

Moreover, we proved that the efficiency score of each point on the Tv′ frontier is α . So, we will take a step towards the Tv′ frontier. Choosing different strategies, a specific inefficient DMU may move toward the frontier in different ways. As an example, change the strategy in input (input decreasing), change in output (output increasing) or combination oriented (input decreasing and output increasing) can be used. In this example, because of a single input and output for decision making units, the chosen strategy is a vertical connecting line between DMUM and hyperplane H1 and a vertical connecting line between DMUN and the hyperplane H2 . We know that the perpendicular line is the shortest distance from the Tv′ frontier. Therefore, we will get the equation of the perpendicular line that intersects the hyperplane H1 from M. This perpendicular line can be obtained as follows: L1 = {(x, y)|8y + 5x = 31}. Then, we obtain the intersection of L1 and H1 . This point is M ′ with coordinates (XM ′ , YM ′ ) = (2.191, 2.505). It means that if the coordinates of the point M (3, 2) change to M ′ (2.191, 2.505), its efficiency will increase from θM∗ = 0.500 to θM ′ = 0.800. Similarly, we obtain the equation of the perpendicular line that intersects the hyperplane H2 from N.

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This perpendicular line can be obtained as follows: L2 = {(x, y)|8y + 10x = 89}. Now, we will obtain the intersection of L2 and H2 . This point is N ′ with coordinates (XN ′ , YN ′ ) = (4.451, 5.560). Additionally, if the coordinates of the point N (4.5, 5.5) change to N ′ (4.451, 5.560), its efficiency will increase from θN∗ = 0.7778 to θN ′ = 0.800. 5. Conclusion In recent years, many DEA researchers have studied the sensitivity analysis of efficient and inefficient unit classifications with respect to perturbations of data. This paper is focused on the sensitivity analysis of inefficient DMUs. As said before, after determining the inefficient units in the society under study, it is needed, in some scientific subjects, that all of these units have at least the efficiency score of α . (α is a constant number smaller than 1 which is defined by the manager.) As an example, suppose that people who are working in different places such as schools, universities, hospitals, banks, companies and etc., have a score efficiency. Sometimes, we need, by paying attention to the occupation sensitivity, that all staffs must have at least a score of α . Obviously, those people with an efficiency score less than the least, should come up with the level by themselves. This paper develops a procedure for performing a sensitivity analysis of the inefficient decision making units (DMUs). The procedure yields an exact ‘‘Necessary Change Region’’ which the efficiency score of a specific inefficient DMU changes to a defined efficiency score. The proposed method has some advantages and disadvantages. The most important problem of this method is that, possibly, for every inefficient unit, according to the nature of the problem, a ‘‘Necessary Change Region’’ would not be achieved explicitly. 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