A note on DEA efficiency assessment using ideal point: An improvement of Wang and Luo’s model
Introduction
Data envelopment analysis (DEA), as initiated and developed by Charnes et al. [2], is a data oriented methodology for identifying efficient production frontiers and evaluating the relative efficiency of decision-making units (DMUs) where each DMU is an entity responsible for converting multiple inputs into multiple outputs. DEA is used to establish a best practice group from among a set of observed units and to identify the units that are inefficient when compared to the best practice group. DEA also indicates the magnitude of the inefficiencies and improvements possible for the inefficient units. The main forms of DEA models and their extensions include the CCR and the BCC models [2], [3]; the additive models [4] and the imprecise DEA models [5]. Modifications and extensions are the assurance region models [6], super-efficiency model [7] and the cone ratio models [8], [9]. Stochastic and chance constrained extensions are among [10], [11], [12], [13], [14].
The beauty of ‘pure’ DEA is that it requires no element of judgment, its pronouncements on efficiency/performance are based only on the data. Thus there is considerable interest in trying to modify DEA, retaining as much of this aspect as possible but at the same improving discrimination. Andersen and Petersen [7] modify the CCR model to allow for a ranking of the efficient units. Sexton et al. [15] developed the cross-evaluation matrix and this approach was extended by Doyle and Green [16], both of which are referred to as the cross-efficiency ranking methods. Wu et al. [17] proposed a revised fuzzy DEA by incorporating both the crisp data and fuzzy data so that efficiencies of DMUs in different subsystems can be assessed. A procedure using DEA to rank DMUs of interests is proposed by combining multivariate statistic approaches such as discriminant analysis of ratios [18] and canonical correlation analysis [19]. Relationship between DEA and multi-criteria decision-making is also discussed quite frequently [20], [21], [22], [23]. Recently, Wang and Luo [1] proposed a ranking approach by incorporating DEA into the technique for order preference by similarity to ideal solution (TOPSIS, 1981). The idea of the ideal point is also adopted to develop the “worst practice DEA” [24], where the evaluation of each DMU relative to the NIP treats inputs as outputs and outputs as inputs.
This article focuses on the recent study of DEA and TOPSIS by Wang and Luo [1] that recently appeared in the journal of Applied Mathematics and Computation. The technique for order preference by similarity to ideal solution (TOPSIS) is a classical method first developed by Hwang and Yoon [25], subsequently discussed by many others [26], [27]. TOPSIS is based on the concept that alternatives should be selected that have the shortest distance from the positive ideal solution (PIS) and the farthest distance from the negative ideal solution (NIS), or nadir. The PIS has the best measures over all attributes, while the NIS has the worst measures over all attributes. Wang and Luo [1] made a significant contribution to the literature by showing that the TOPSIS idea can be combined to DEA for a comprehensive ranking of DMUs. However, this paper finds that their method is problematic in employing the negative ideal point (NIP) for DEA computation. The two DEA models using the positive ideal point (PIP) and the negative ideal point (NIP) are logically conflicted. Thus the rational of the approach is naturally doubtable.
In this paper, we aim to disclose that it is problematic to use radial efficiency score as “distances” in Wang and Luo’s [1] approach. We show that it is possible to interpret their NIP based model (7) as a “standard” input-oriented [7] model, if inputs are treated as outputs and outputs are treated as inputs. Thus the PIP measures rely on an input orientation while the NIP measures rely on an output orientation. As is well known, the Farell measure of efficiency corresponds to both the inverse of Shepards’s distance function and the CCR DEA measure of efficiency except the impact from the non-Archimedian lower bounds on the multipliers. However, the Shepards’s distance function is not really a measure of distance (a norm based on a metric). Hence, it is problematic to add differences of output-oriented and input-oriented Farell measures, as proposed in [1]. Realizing this problem, we slightly revise the approach to determine the worst efficiency of the NIP so that it is logically acceptable for using both the PIP and the NIP.
We begin in the following section with a re-examination of Wang and Luo’s Model. Section 3 provides a model improvement. Section 4 gives the numerical illustration. Finally, our conclusions and discussion are presented in Section 5.
Section snippets
A re-examination of Wang and Luo’s model
Consider n DMUs to be evaluated, DMUj (j = 1, 2, … , n) consumes the amounts Xj = {xij} of m different of inputs (i = 1, 2, … , m) and produces the amounts Yj = {yrj} of r outputs (r = 1, … , s). The positive ideal point (PIP) and negative ideal point (NIP), which is referred to as the ideal DMU (IDMU) and anti-ideal DMU (ADMU) by Wang and Luo’s [1] are respectively defined asThis definition means that a PIP (IDMU)
A model improvement
Although we argue that the lack of discrimination power is not really a problem of DEA as those entities located on the efficient frontier are essentially equal according to the available data, however, in the MCDM context, concerned with the prospective performance of the entities under consideration for the purpose of choosing the non-dominated entity, many efficiency alternatives is problematic.
In that sense, we hereby revise Wang and Luo’s [1] models so that a DEA analysis using TOPSIS idea
Numerical illustration
Numerical illustration is done regarding the examples in [1] where two examples are given. Result of example 1 using our proposed method is documented in Table 3. This result is compared with that in [7] where a ranking approach is suggested to compare the DMU with a linear combination of all the other DMUs, i.e., the DMU under evaluation itself is excluded. The ranking order of Andersen and Petersen [7] was: DMU2 ≻ DMU4 ≻ DMU3 ≻ DMU1 ≻ DMU5, where the symbol “≻” denotes “is superior to”. Our result
Conclusions and discussion
The main objective of this note has been to highlight the drawbacks in the approach by Wang and Luo [1] and to present a revised approach so that DEA efficiency assessment based on ideal points can be performed. Wang and Luo [1] contribute to a very interesting issue by showing that the TOPSIS idea can be combined to DEA for a comprehensive ranking of DMUs. However, their approach is problematic in employing the negative ideal point (NIP) for DEA computation. The ideal point based models rely
Acknowledgement
The author expresses deep appreciation to the anonymous reviewers for their constructive comments and suggestions that improved the presentation of this paper.
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