Zero weights and non-zero slacks: Different solutions to the same problem

Abstract

This paper re-assesses three independently developed approaches that are aimed at solving the problem of zero-weights or non-zero slacks in Data Envelopment Analysis (DEA). The methods are weights restricted, non-radial and extended facet DEA models. Weights restricted DEA models are dual to envelopment DEA models with restrictions on the dual variables (DEA weights) aimed at avoiding zero values for those weights; nonradial DEA models are envelopment models which avoid non-zero slacks in the input-output constraints. Finally, extended facet DEA models recognize that only projections on facets of full dimension correspond to well defined rates of substitution/transformation between all inputs/outputs which in turn correspond to non-zero weights in the multiplier version of the DEA model. We demonstrate how these methods are equivalent, not only in their aim but also in the solutions they yield. In addition, we show that the aforementioned methods modify the production frontier by extending existing facets or creating unobserved facets. Further we propose a new approach that uses weight restrictions to extend existing facets. This approach has some advantages in computational terms, because extended facet models normally make use of mixed integer programming models, which are computationally demanding.

Families of linear efficiency programs based on Debreu’s loss function

“…The first concerns relations among programs. Portela and Thanassoulis (2006) claim that, under certain circumstances, weight restriction programs and non-radial programs are equivalent. This is a very strong claim.…”

Assessing the performance of alternative investments using non-parametric efficiency measurement approaches: Is it convincing?

“…Following Portela and Thanassoulis (2006), we therefore consider the inclusion of value judgements to restrict the number of zero weights. Several approaches to incorporate value judgements have been suggested in the literature, comprising -among othersabsolute and relative weights restrictions.…”

“…18 The input-oriented Russell measure is basically equivalent to the input-oriented slacks-based measure proposed by Tone (2001). 19 See Portela and Thanassoulis (2006). 20 Even if we restrict only the marginal rates of substitution, we obtain different results because the input-oriented Russell measure sets the lower bound for each m = 1, .…”

A multiplier bound approach to assess relative efficiency in DEA without slacks

“…In this example, this would be the segment BC, and so DMU D would be again projected onto H C with the same results as before. Portela and Thanassoulis (2006) determine a lower and an upper bound for the ratios between each couple of multipliers as the minimum and maximum of the corresponding marginal rates of substitution estimated with all the FDEFs of the frontier. The resulting AR-model for the assessment of DMU D would have as feasible solutions those associated with all the supporting hyperplanes for the PPS at DMU B which are in between H A and H C .…”

“…Some of these approaches are based on the extension of full dimensional efficient facets (FDEFs) of the frontier. See Green et al (1996), Olesen and Petersen (1996) and Portela and Thanassoulis (2006). In the two former, the authors define some new technologies from the FDEFs of the original frontier, whereas in the latter the proposed approach uses an AR model in which the bounds of the AR constraints are estimated from the marginal rates of substitution of these FDEFs.…”