On VEA, production trade-offs and weights restrictions

Abstract

In this article we explore the relationship between Value Efficiency Analysis (VEA) and Data Envelopment Analysis (DEA) models including production trade-offs or weights restrictions. In particular, we show that the VEA model is equivalent to a DEA model including production trade-offs for which the trade-off coefficient vectors are equal to either (i) the negative of the input and output quantities of the Decision Making Units (DMUs) chosen as the Most Preferred Solution (MPS) in VEA, under constant returns to scale, or (ii) the deviations of all evaluated DMUs’ input and output quantities from those of the DMUs chosen as the MPS, irrespectively of the returns-to-scale assumption. These trade-offs are the dual forms of type II Assurance Region weight restrictions. We then show that a similar equivalence holds between pure output or input VEA and DEA models including trade-offs, if the above trade-offs are respectively considered only for outputs or only for inputs. In this case the trade-offs are the dual forms of type I Assurance Region weight restrictions.

Keywords:

• Data envelopment analysis
• value efficiency analysis
• weight restrictions
• production trade-offs

Acknowledgements

We would like to thank the Editor and two anonymous reviewers for constructive comments and suggestions. An earlier version of this article was presented in the XVI European Workshop on Efficiency and Productivity Analysis (EWEPA), held at London, UK, June 10–13, 2019. We would also like to thank the session’s participants for a fruitful discussion and comments.

Disclosure statement

The authors report no relevant financial or non-financial competing interests.

Notes

1 This is also one of the main reasons motivating the incorporation of weights restrictions

2 We focus on the input-oriented model, but our results can be straightforwardly extended to the output-oriented model.

3 Podinovski (2004) has shown that non-homogeneous linear weight restrictions, i.e., those for which the right-hand side of (4)(4) $$\sum _{j=1}^J{u}_j^k{q}_j^r-\sum _{i=1}^I{v}_i^k{p}_i^r\le 0,\mathrm{ }r=1,\dots ,\mathrm{ }R\mathrm{ }$$(4) is non-zero, can also be represented in the form of production trade-offs in the envelopment form of the DEA model.

4 Cases (i) and (ii) may also refer to a subset of inputs and outputs if respectively $p_i^r=0$ for some i in Case (i) and $q_j^r=0$ for some j in Case (ii).

5 In Charnes et al. (1991) the DMUs with a DEA efficiency score of one are classified into three categories: (i) extreme-efficient DMUs that reside at a point of the convex DEA frontier where more than one facets intercept, (ii) non-extreme-efficient DMUs, namely DMUs located on the interior of a facet, and (iii) weakly-efficient DMUs that have at least one positive optimal value for an input or output slack. An MPS chosen by the DMUs in (ii) can be expressed as a linear combination of DMUs in (i). If the DM chooses a dominated (i.e., DEA-inefficient or weakly-efficient) DMU as the MPS, then his/her DM preferences can be stated equivalently by using as the MPS the combination of the extreme-efficient DMUs that are identified as peers of the chosen DMU, as the MPS (see Halme et al., 1999).

6 Weight restrictions that result in extending facets of the DEA frontier are discussed in Portela and Thanassoulis (2006), but are not related to VEA.

7 Note that when we consider only outputs it makes no sense to have an input-oriented model. Also, as Lovell and Pastor (1999) have shown, a pure-output CRS output-oriented DEA model rates all DMUs as infinitely inefficient, while an input-oriented VRS DEA model with a single constant input rates all DMUs as efficient.

8 Variants of (19)(19) $$\begin{array}{ll}\begin{array}{l}\mathrm{ }\underset{u_j^k,\mathrm{ }{v}_i^k}{\max }\mathrm{ }\sum _{j=1}^J{u}_j^k{y}_j^k\\ \mathrm{ }s.t.\mathrm{ }\sum _{j=1}^J{u}_j^k{y}_j^h\le 1\mathrm{\hspace{1em}}h=1,\dots ,K\\ \begin{array}{l}\mathrm{ }{u}_j^k\ge 0\mathrm{\hspace{1em}}j\mathrm{ }=1,\dots ,\mathrm{ }J\end{array}\end{array}& \begin{array}{l}\underset{{\lambda }_h^k}{\min }\sum _{h=1}^K{\lambda }_h^k\\ \mathrm{ }s.t.\mathrm{ }\sum _{h=1}^K{\lambda }_h^k{y}_j^h\ge y_j^k\mathrm{\hspace{1em}}j=1,\dots ,J\mathrm{ }\\ \mathrm{ }\begin{array}{l}\mathrm{ }{\lambda }_h^k\ge 0\mathrm{\hspace{1em}}h=1,\dots ,K\\ \mathrm{ }{\theta }_{}^k\mathrm{ }\mathit{free}\end{array}\end{array}\end{array}$$(19) including weight restrictions have been employed in, among others, the construction of composite indicators of environmental performance (Zanella et al., 2013), the re-estimation of the Technology Achievement Index (Cherchye et al., 2008), and the aggregation of several measures of money into a synthetic indicator (Sahoo & Acharya, 2010).

9 Note that when we consider only inputs it makes no sense to have an output-oriented model. Also, as Lovell and Pastor (1999) have shown, a pure-input CRS input-oriented DEA model rates all DMUs as infinitely inefficient, while an output-oriented VRS DEA model with a single constant output rates all DMUs as efficient.

10 Variants of (21)(21) $$\begin{array}{ll}\begin{array}{l}\mathrm{ }\underset{v_i^k}{\min }\mathrm{ }\sum _{i=1}^I{v}_i^k{x}_i^k\\ \mathrm{ }s.t.\mathrm{ }\sum _{i=1}^I{v}_i^k{x}_i^h\ge 1\mathrm{ }h=1,\dots ,K\\ \begin{array}{l}\mathrm{ }{v}_i^k\ge 0\mathrm{\hspace{1em}}i\mathrm{ }=1,\dots ,I\end{array}\end{array}& \begin{array}{l}\underset{{\lambda }_h^k}{\max }\sum _{h=1}^K{\lambda }_h^k\\ \mathrm{ }s.t.\mathrm{ }\sum _{h=1}^K{\lambda }_h^k{x}_i^h\le x_i^k\mathrm{\hspace{1em}}i=1,\dots ,I\mathrm{ }\\ \mathrm{ }\begin{array}{l}\mathrm{ }{\lambda }_h^k\ge 0\mathrm{\hspace{1em}}h=1,\dots ,K\end{array}\end{array}\mathrm{ }\end{array}$$(21) including weight restrictions have been used by, among others, Zhou et al. (2007) to construct a sustainable energy index, and Rogge (2012) to re-estimate the Environmental Performance Index.

11 The applications of DEA and other multi-criteria decision-making methods in technology selection are nowadays voluminous and include, but are not limited to, the selection of flexible manufacturing systems, industrial robots, and dispatching rules. A review of such applications is a task out of the scope of the present article, and the interested reader is referred to Hamzeh and Xu (2019), for a recent review.

Additional information

Funding

This work was supported by the Hellenic Foundation for Research and Innovation (HFRI) and the General Secretariat for Research and Technology (GSRT) under Grant 698.