Using the gradient line for ranking DMUs in DEA
Introduction
Data envelopment analysis (DEA) proposed by Cooper and co-workers [3], and further extended by Banker et al. [2], and others, is used to evaluate the relative efficiency of decision making units (DMUs). When DEA models are used to calculate efficiency of DMUs a number of them may have an equal to efficiency one. To rank these DMUs, two methods are proposed by Andersen and Petersen [1] called AP method, and another method proposed by Mehrabian et al. [4] called MAJ method. These methods would fail if data have special structure. The method proposed by the authors fully ranks such DMUs. The paper contains seven sections. In Section 2 some preliminary knowledge of DEA is put forward. Section 3 contains AP and MAJ models. In Section 4 equation of gradient line is discussed. The proposed method is introduced in Section 5. Some numerical examples are solved in Section 6. Finally, in Section 7 the conclusion and some remarks are put forward.
Section snippets
Production possibility set
Consider observed output Yj=(y1j,…,ysj)⩾0 and input Xj=(x1j,…,xmj)⩾0, Xj≠0, Yj≠0 for each of j=1,…,n DMUs. The DEA postulates that underlying the production possibility set (PPS) possess the following properties: Postulate 1 Nonempty The observed (Xj,Yj)∈T, j=1,…,n. Postulate 2 Constant returns to scale If (X,Y)∈T, then (λX,λY)∈T for all λ⩾0. Postulate 3 Convexity T is a closed and convex set, i.e. if (X1,Y1)∈T and (X2,Y2)∈T then for λ∈(0,1), λ(X1,Y1)+(1−λ)(X2,Y2)∈T. Postulate 4 Plausibility If (X,Y)∈T, Xt⩾X and Yt⩽Y, then (Xt,Yt)∈T. Postulate 5 Minimum extrapolation T is the
AP model
This model was proposed by Andersen and Petersen [1] for ranking efficient units, as follows:
AP model, in some cases, breaks down with zero data and may be unstable because of extreme sensitivity to small variations in the data when some DMUs have relatively small values for some of its inputs. These cases are discussed in detail in [7]. Example 1 Consider three DMUs with two
Equation of the gradient line (ellipse)
Consider DMUp with inputs and outputs equal to Xp=(xip,i=1,…,m) and Yp=(yrp,r=1,…,m), respectively. Let P0 be two-dimensional plane (which is called (α,β) space) floated into (Xp,Yp) and a set S0, its intercept with the production set, as proposed in Hackman et al. [6]:
The equation of the gradient line (ellipse) corresponding of DMUp iswhereandFor detail see [5].
Using the gradient line for ranking DMUs
Consider the following Additive model with constant return to scale [8]:By multiplying objective function with ϵ>0 we haveThe dual of this model will bewhere, V is m-vector, U is s-vector and T is a
Illustrative examples
Example 3 Consider the same data in Example 1. The following table shows the results: As can be seen, the corresponding problem of DMU2 in the AP and MAJ models are infeasible. So these models cannot rank all extreme efficient DMUs. But the proposed method ranks DMU2, DMU3 and DMU1 which their rank are 1, 2 and 3, respectively. Example 4 Consider 15 DMUs with four inputs and three outputs.DMUj Result Length of the arc New method AP MAJ 1 0.18 Eff 0.223361 3 2 2 2 1 Eff 1.253439 1 Infeasible Infeasible 3 0.33 Eff 0.496267 2 1 1 DMUj I1 I2 I3 I4 O1
Conclusion
In this paper, a method is proposed for ranking the efficient DMUs. The suggested ranking method ranks the DMUs with the use of the gradient line. Since the existing models are not proper when data have special structure, the method proposed fully ranks such DMUs. In this method, the model is always feasible and it is independent from orientation. The method suggested removes the difficulties confronted by using the existing models for ranking DMUs.
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Cited by (12)
Review of efficiency ranking methods in data envelopment analysis
2017, Measurement: Journal of the International Measurement ConfederationCitation Excerpt :This model overcome the infeasibility issue of Anderson and Peterson [12] model plus the sensitivity of their model to small variation in data when the DMUs have smaller values as inputs or outputs. In the referred papers [23,56,57] Jahanshahloo et al. proposed some different models as alternated developed versions of the MAJ and AP models. With similar goal of overcoming infeasibility of the super efficiency model, Aldamak et al. [58] proposed different method of calculating efficiency when convexity assumption is relaxed by using free disposal hull (FDH) approach.
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