Available online at www.ispacs.com/dea Volume 2014, Year 2014 Article ID dea-00069, 5 Pages
doi:10.5899/2014/dea-00069 Comment
A comment on “ranking efficient DMUs based on a single
virtual inefficient DMU in DEA”
Mohammad Izadikhah∗
Department of Mathematics, College of Science,Arak-Branch, Islamic Azad University, Arak, Iran
Copyright 2014 c⃝ Mohammad Izadikhah. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
In a recent paper by Shetty and Pakkala [U. Shetty, T.P.M. Pakkala, (2010), Ranking efficient DMUs based on single virtual inefficient DMU in DEA. OPSEARCH, 47 (1):50-72], they proposed an approach to rank the efficient decision making units based on a virtual DMU. The input and output levels of this virtual DMU are the average of input and outputs of all DMUs. They showed that, if there exist at least one inefficient DMU then this virtual DMU is inefficient. Also, based on this virtual DMU they proposed a model for ranking efficient DMUs. This brief comment provides an alternative analytical and direct proof for their main theorem.
Keywords:Data envelopment analysis; Decision making units; Virtual DMU; Domination.
1 Introduction
Data envelopment analysis (DEA) is a non-parametric linear programming method capable of the efficiency evalu-ation of decision making units. DEA first proposed by Charnes et al. [4], in evaluating the efficiency of an educevalu-ational center in USA. However, the original DEA method does not differentiate the efficient firms and thus, does not create full ranking. To overcome this problem, several methods have been developed with the aim of enlarging the dis-tinguishing power of DEA. Several authors have proposed methods for ranking the best performers. First, in 1985, Charnes et al. [3], counted the number of times that an efficient DMU play the role of benchmark unit for others, and used this norm to rank these units. Since, finding the reference set of a DMU is not easy, their model is not a suitable method. Sexton et al. [15], in 1986 suggested the cross efficiency method. The super efficiency method for ranking DMUs, first proposed by Andersen and Petersen [2] in 1993, the super-efficiency technique ranks DMUs through the exclusion of the unit being scored from both the DEA dual linear program and the reference set. Since the AP-model can, in practical application, lead to infeasible programs when some of the inputs vanish or large efficiency score when some of the inputs are small therefore in 1999 Mehrabian et al. [13]proposed the MAJ-model that alleviates these problems. Tone [17] in 2002 proposed a slacks-based measure of super-efficiency in data envelopment analysis. For a review of ranking methods before 2002, readers are refereed to Adler et al. [1]. In 2004, Jahanshahloo et al. [12] presented a method for ranking extreme efficient decision making units in data envelopment analysis models with constant and variable returns to scale. In their method, they exploit the leave-one-out idea and l1-norm. In 2006, Ja-hanshahloo and Afzalinejad [9] proposed a method for ranking DMUs. In their method, DMUs are compared against an full-inefficient frontier. In 2007, Jahanshahloo et al. [10] proposed a new ranking system for extreme efficient
DMUs based upon the omission of the efficient DMUs from reference set of the inefficient DMUs. They stated and proved some facts related to their model. For a review of ranking methods before 2008, readers are refereed to Jahan-shahloo et al. [11]. In 2010, Du et al. [6] proposed a new DEA-based method for fully ranking all decision-making units. To improve the method proposed by Jahanshahloo et al. [10] this method was based on the combination of each efficient DMUs influence on all the other DMUs and the standard efficiency scores. In the same year, Shetty and Pakalla [16] Proposed a method for ranking efficient DMUs based on a single virtual inefficient DMU in DEA. Again in 2010, Hosseinzadeh Lotfi et al. [8] proposed an algorithm on DEA in which different layers of efficiency were used to rank the efficient DMUs. In 2012, Xu and Dan [18] introduced two alternative efficiency measures by using efficient and anti-efficient frontiers in DEA and proposed a new ranking system for all DMUs. In the same year, Rezai Balf et al. [14] proposed a method for ranking extreme efficient decision making units. Their method used L∞ (or Tchebycheff) Norm, and it seems to have some superiority over other existing methods, because their method was able to remove the existing difficulties in some methods, such as Andersen and Petersen [2] (AP) that it is sometimes infeasible. In 2013, Chen et al. [5] proposed a super-efficiency based on a modified directional distance function. The aim of their method was to modify the directional distance function by selecting proper feasible reference bundles so that the resulting NL measure of super-efficiency is always feasible. In the same year, Gholam Abri et al. [7] proposed a method for ranking non-extreme efficient units.
In a recent paper in this journal, Shetty and Pakkala [16] proposed an approach to rank the efficient decision making units based on a virtual DMU. The input and output levels of the virtual DMU are the average of inputs and outputs of all DMUs. They showed that, if there exist at least one inefficient DMU then this virtual DMU is inefficient. Their proof may seems not so analytical. However, in this comment an alternative analytical and direct proof for their main theorem is provided.
Rest of the paper is organized as follows: In section two, we review the Shetty and Pakkala method. In section three, we will focus on the alternative proof of the Shetty and Pakkala’s main theorem. And finally, the conclusion is discussed in Section four.
2 Shetty and Pakkala method
Assume that there are n DMUs, where each DMUj(j=1,. . . ,n), uses m different inputs, xi j(i=1,. . . ,m), to produce
s different outputs, yr j(r=1,. . . ,s). We assume that the data set are positive. Virtual DMU is denoted by (xAv, yAv) such
that: xiAv=1n n
∑
j=1 xi j, i=1,. . . ,m; yrAv=1n n∑
j=1 yr j, r=1,. . . ,s.Their approach proposed a measure to discriminate efficient units. This is achieved by measuring the efficiency of the virtual DMU by deleting efficient DMUs one by one. For this purpose they proposed the following model:
δAv,b= max s
∑
r=1 uryrAv s.t. m∑
i=1 vixiAv= 1, s∑
r=1 uryr j− m∑
i=1 vixi j≤ 0, j∈ J − b s∑
r=1 uryrAv− m∑
i=1 vixiAv≤ 0, ur≥ε, vi≥ε, ∀r,i. (2.1)Where J ={1,...,n} is the set of DMUs, b ∈ E where E is set of CRS efficient DMUs. They showed that, if there exists at least one inefficient DMU then this virtual DMU is inefficient. However, an alternative analytical and direct
3 Alternative proof of the main theorem
First, we consider the following definition of domination.
Definition 3.1. Let DMUA= (xA, yA) and DMUB= (xB, yB) be two arbitrary units. Then we say DMUAdominates
DMUBif xA≤ xBand yA≥ yBand strictly inequality holds for at least one component.
In this paper, the efficiency score of DMUAis denoted byθACCRwhen is evaluated by CCR model. Now, we prove the
following theorem.
Theorem 3.1. The virtual DMU is inefficient if there be at least one inefficient DMU in the set.
Proof. Suppose that DMUkbe an inefficient DMU, therefore in evaluating DMUkwith CCR model we haveθkCCR< 1.
Thus in optimality there exists a nonnegative vectorλ∗= (λ1∗, . . . ,λn∗) such that: n
∑
j=1 λ∗ jxi j≤θkCCRxik< xik, i=1,. . . ,m; n∑
j=1 λ∗ jyr j≥ yik, r=1,. . . ,s.Where, (θkCCR,λ∗) is the optimal solution of CCR model in evaluating DMUk. It is evident that in this case all of the
input constraints are in the form of strictly inequalities, and since
xiAv=1n n
∑
j=1 xi j, i=1,. . . ,m; yrAv=1n n∑
j=1 yr j, r=1,. . . ,s.thus for input constraints we have:
n
∑
j=1 λ∗ jxi j< xik, ⇒ n∑
j=1, j̸=k xi j+ n∑
j=1 λj∗xi j< n∑
j=1, j̸=k xi j+ xik= n∑
j=1 xi j, ⇒ 1 n( n∑
j=1, j̸=k xi j+ n∑
j=1 λ∗ jxi j) < 1 n n∑
j=1 xi j= xiAv, ⇒ n∑
j=1, j̸=k (1 +λ ∗ j n )xi j+ λ∗ k n xik< xiAv.Similarly, for output constraints we have:
n
∑
j=1, j̸=k (1 +λ ∗ j n )yr j+ λ∗ k n yrk≥ yrAv, r=1,. . . ,s;Therefore, there exists a nonnegative vector ˆλ = ( ˆλ1, . . . , ˆλn) such that ( n
∑
j=1 ˆ λjxj, n∑
j=1 ˆ λjyj) dominates (xAv, yAv), and ˆ λj= 1+λ∗j n , j=1,. . . ,n, j̸= k; ˆ λk=λ ∗ k n, .Specially, the following relations are hold: n
∑
j=1 ˆ λjxi j< xiAv, i=1,. . . ,m; n∑
j=1 ˆ λjyr j≥ yrAv, r=1,. . . ,s.Thus, there existsα< 1 such that for input constraints we have:
n
∑
j=1
ˆ
λjxi j=αxiAv, i=1,. . . ,m;
Now, if we evaluate (xAv, yAv) with CCR model then (α, ˆλ) will be a feasible solution for that model. Since the CCR
model is in the minimization form, therefore we have:θAvCCR≤α< 1. So, (xAv, yAv) is inefficient.
4 Conclusions
Lack of discrimination power is a drawback of DEA that has aroused considerable research interest in the DEA literature. And in many cases, it is necessary to give a full ranking of the DMUs. For this purpose, different methods with different properties to achieve full ranking have been proposed. Recently, Shetty and Pakkala proposed a method for ranking efficient DMUs based on single virtual inefficient DMU in DEA. Therefore, in this paper an alternative analytical and direct proof for Shetty and Pakkala’s main theorem is provided.
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