Universal efficiency scores in data envelopment analysis based on a robust approach

Abstract

We propose a novel DEA ranking based on a robust optimization viewpoint: the higher ranking for those DMU's that remain efficient even for larger simultaneous and independent variations of all data and vice versa. This ranking can be computed by solving generalized linear fractional programming problems, but we also present a tight linear programming approximation that preserves the order of rankings. We show many remarkable properties of our approach: It preserves the order of rankings compared to the classical approach, and it is unit invariant. It is naturally normalized, so it can be used as universal ranking of DMU's of unrelated models. It gives ranking not only for inefficient, but also for efficient decision making units. It can also be easily extended to generalized or alternative models, for instance to deal with interval data. We present several examples confirming the desirable properties of the method.

The Impact of Operating Environment on Efficiency of Public Libraries

“…Many more models are proposed in the literature addressing various issues in DEA. A particulary convenient and elegant model is the Chebyshev distance model of Hladík (2019). It is based on the robust optimization viewpoint and has many attractive properties such as the super-efficiency, i.e.…”

“…To obtain technical efficiencies, we utilize the Chebyshev distance DEA with variable returns to scale (VRS) proposed by Hladík (2019). Let X = (x i,j ) n,r i=1,k=1 be the non-negative matrix of inputs and Y = (y i,k ) n,s i=1,k=1 be the non-negative matrix of outputs.…”

“…, ν i,s ) are the weights of outputs and ϕ i is the auxiliary variable used for ensuring VRS. The above formulation is a non-linear optimization problem which Hladík (2019) further propose to linearize. Let us reparametrize the weights and the VRS variable as νi =…”

“…(3) Hladík (2019) shows in several examples that the linear approximation ( 3) is quite precise and can be effectively utilized in practice.…”

“…DEA is a non-parametric method measuring how efficiently can decision making units (DMU) transform a set of inputs to a set of outputs. We utilize the Chebyshev distance DEA model with variable returns to scale recently proposed by Hladík (2019). This model is based on the robust optimization viewpoint and has many desirable properties -super-efficiency, comparability of efficiency scores across different analyzes, inclusion of zero inputs and outputs, units invariance, order of rankings identical to the classical approach and straightforward interpretability.…”