On characterizing full dimensional weak facets in DEA with variable returns to scale technology

Abstract

The frontier of the Production Possibility Set (PPS) consists of two types of full dimensional facets, efficient and weak facets. Identification of all facets of the PPS can be used in sensitivity and stability analysis, to find the closet target for inefficient Decision-Making Units (DMUs), and to determine the status of returns to scale of a DMU, among others. There has been a surge of articles on determining efficient facets in recent years. There are, however, many cases where knowledge of weak facets is required for a thorough analysis. This is the case, in particular, when the frontier of the PPS is constructed only of weak facets. The existing algorithms for finding weak facets either require knowledge of all extreme directions of the PPS or applicable only under some restrictions on the position of weak efficient DMUs. We provide a complete characterization of weak facets. Using this characterization, we then devise a different algorithm to find weak facets. We illustrate our algorithm using a numerical example.

Keywords:

• data envelopment analysis
• full dimensional weak facet
• mixed integer linear programming
• polyhedral sets

AMS Subject Classifications:

• 90B50
• 90C05
• 90C11

Acknowledgments

The authors thank the referees for their constructive and insightful comments. The research of Masoud Asgharian and Vahid Partovi Nia is partly supported by NSERC of Canada.

Notes

1 It is verified that generating all vertices of an unbounded polyhedron is NP-hard, whereas in the case of bounded polyhedral, the complexity is still open.

2 It can be shown that the set of optimal solutions of model (55 ) for some extreme BCC-efficient DMUs is an unbounded polyhedron thereby the computational complexity of finding all FDWFs is NP-hard.