Using the gradient line for ranking DMUs in DEA

doi:10.1016/S0096-3003(03)00333-3

Applied Mathematics and Computation

Volume 151, Issue 1, 30 March 2004, Pages 209-219

https://doi.org/10.1016/S0096-3003(03)00333-3 Get rights and content

Abstract

In this paper a method, based on using gradient line, for ranking DMUs is proposed. The advantage of this method is its stability and robustness, where data have special structure. Some numerical examples are presented to show the results.

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 Y_j=(y_1j,…,y_sj)⩾0 and input X_j=(x_1j,…,x_mj)⩾0, X_j≠0, Y_j≠0 for each of j=1,…,n DMUs. The DEA postulates that underlying the production possibility set (PPS) $T={(X,Y)| output vector Y⩾0 can be produced from input vector X⩾0}$ possess the following properties:

Postulate 1 Nonempty

The observed (X_j,Y_j)∈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, X_t⩾X and Y_t⩽Y, then (X_t,Y_t)∈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: $Min θ_{p} −ϵ ∑_{i=1}^{m} s_{i}^{−} +∑_{r=1}^{s} s_{r}^{+},$ $s . t . ∑_{j=1j≠p}^{n} λ_{j} x_{ij} +s_{i}^{−} =θ_{p} x_{ip}, i=1,…,m,$ $∑_{j=1j≠p}^{n} λ_{j} y_{rj} −s_{r}^{+} =y_{rp}, r=1,…,s,$ $λ_{j} ⩾0, j=1,…,n,$ $s_{i}^{−} ⩾0, i=1,…,m,$ $s_{r}^{+} ⩾0, r=1,…,s.$

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 DMU_p with inputs and outputs equal to X_p=(x_ip,i=1,…,m) and Y_p=(y_rp,r=1,…,m), respectively. Let P₀ be two-dimensional plane (which is called (α,β) space) floated into (X_p,Y_p) and a set S₀, its intercept with the production set, as proposed in Hackman et al. [6]: $P_{0} ={(X,Y)|X=αX_{p},Y=βY_{p}},$ $S_{0} ={(X,Y)∈P_{0} |X=αX_{p},Y=βY_{p}, α⩾0, β⩾0}.$

The equation of the gradient line (ellipse) corresponding of DMU_p is $α^{2} K_{α}^{2} + β^{2} K_{β}^{2} =1,$ where $K_{α} = ∑_{i=1}^{m} x_{ip}^{2} +∑_{r=1}^{s} y_{rp}^{2} ∑_{i=1}^{m} x_{ip}^{2}$ and $K_{β} = ∑_{i=1}^{m} x_{ip}^{2} +∑_{r=1}^{s} y_{rp}^{2} ∑_{r=1}^{s} y_{rp}^{2} .$ For detail see [5].

Using the gradient line for ranking DMUs

Consider the following Additive model with constant return to scale [8]: $Min − ∑_{i=1}^{m} s_{i}^{−} +∑_{r=1}^{s} s_{r}^{+},$ $s . t . −∑_{j=1}^{n} λ_{j} x_{ij} −s_{i}^{−} =−x_{ip}, i=1,…,m,$ $∑_{j=1}^{n} λ_{j} y_{rj} −s_{r}^{+} =y_{rp}, r=1,…,s,$ $λ_{j} ⩾0, j=1,…,n,$ $s_{i}^{−} ⩾0, i=1,…,m,$ $s_{r}^{+} ⩾0, r=1,…,s.$ By multiplying objective function with ϵ>0 we have $Min −ϵ ∑_{i=1}^{m} s_{i}^{−} +∑_{r=1}^{s} s_{r}^{+},$ $s . t . −∑_{j=1}^{n} λ_{j} x_{ij} −s_{i}^{−} =−x_{ip}, i=1,…,m,$ $∑_{j=1}^{n} λ_{j} y_{rj} −s_{r}^{+} =y_{rp}, r=1,…,s,$ $λ_{j} ⩾0, j=1,…,n,$ $s_{i}^{−} ⩾0, i=1,…,m,$ $s_{r}^{+} ⩾0, r=1,…,s.$ The dual of this model will be $Max H_{p} =−V^{T} X_{p} +U^{T} Y_{p},$ $s . t . −V^{T} X_{j} +U^{T} Y_{j} ⩽0, j=1,…,n,$ $V⩾ϵ 1,$ $U⩾ϵ 1 .$ where, 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:

DMU_j	$H_{j}$	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

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.

DMU_j

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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S. Mehrabian et al.
A complete efficiency ranking of decision making units: an application to the teacher training university
Computational Optimization and Application
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There are more references available in the full text version of this article.

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View full text

Using the gradient line for ranking DMUs in DEA

Abstract

Introduction

Section snippets

Production possibility set

AP model

Equation of the gradient line (ellipse)

Using the gradient line for ranking DMUs

Illustrative examples

Conclusion

European Journal of Operational Research

A procedure for ranking efficient units in data envelopment analysis

Management Science

Some methods for estimating technical and scale inefficiencies in data envelopment analysis

Management Science

A complete efficiency ranking of decision making units: an application to the teacher training university

Computational Optimization and Application