Adding and removing an attribute in a DEA model: theory and processing

Abstract

We present a theoretical and computational study of the impact of inserting a new attribute and removing an old attribute in a data envelopment analysis (DEA) model. Our objective is to obviate a portion of the computational effort needed to process such model changes by studying how the efficient/inefficient status of decision-making units (DMUs) is affected. Reducing computational efforts is important since DEA is known to be computationally intensive, especially in large-scale applications. We present a comprehensive theoretical study of the impact of attribute insertion and removal in DEA models, which includes sufficient conditions for identifying efficient DMUs when an attribute is added and inefficient DMUs when an attribute is removed. We also introduce a new procedure, HyperClimb, specially designed to quickly identify some of the new efficient DMUs, without involving LPs, when the model changes with the addition of an attribute. We report on results from computational tests designed to assess this procedure's effectiveness.

Appendix

Result 1

• A data point that is on the boundary of the production possibility set in dimension m is on the boundary of the production possibility set in dimension m+1.

Proof

• We present the demonstration for the VRS model and then explain how to adapt it for the other returns to scale assumptions.

  Let 𝒜 be a set of n data points in ℛ^m. Assume that a ^j*∈𝒜 is on the boundary of the production possibility set generated by the VRS model in dimension m. Then, there exist 0≠π ^*∈ℛ^m, π ^*⩾0, and β∈ℛ such that

  

  Thus, for â^j*∈ℛ^m+1 and π _m+1 ^*=0,

  

  Then, â^j* is on the boundary of the production possibility set in dimension m+1. □

  The proofs for the IRS and the DRS follow immediately by adding above the constraint β⩾0, or β⩽0, respectively. For the CRS, the demonstration is as above with β=0.

Result 2

• An extreme point of the production possibility set in ℛ^m is extreme in ℛ^m+1.

Proof

• Let be an extreme point of the production possibility set in . Then there exists a supporting hyperplane such that is the only element of the support set. That is:

  

  Construct the vector . Then

  

  This means that the hyperplane in supports the higher-dimensional VRS production possibility set (with support set an unbounded face). The point is necessarily extreme since it is the only point in that face. Since is an extreme point of this face it is an extreme point of the full polyhedral set. □

  The proofs for the IRS, the DRS, and the CRS follow immediately by adding above the constraints β⩾0, β⩽0, or β=0 respectively.

Result 4

• Let a ^j*∈𝒜 be a boundary element of the production possibility set. Assume that there exists a hyperplane that supports the hull at a ^j* such that, w.l.o.g., is the only coordinate of that is zero. Then a ^j* is efficient when removing the mth attribute from the DEA model. If, in addition a ^j*, is the only element of the support set in dimension m, then it is extreme efficient in the lower dimension.

Proof

• Since supports the hull at a ^j* if follows that

  

  This implies that

  

  Hence

  

  Since , it follows that a ^j* if efficient when the mth attribute is removed from the DEA model.

  It is obvious that if a ^j* is the only element of the support set in dimension m, then it is extreme efficient when removing the mth attribute. □

The proofs for the IRS, the DRS, and the CRS follow immediately by adding above the constraints β⩾0, β⩽0, or β=0 respectively.