Handling negative data in slacks-based measure data envelopment analysis models

doi:10.1016/j.ejor.2019.09.055

European Journal of Operational Research

Volume 282, Issue 3, 1 May 2020, Pages 926-935

https://doi.org/10.1016/j.ejor.2019.09.055 Get rights and content

Highlights

•
We develop slacks-based data envelopment analysis models to deal with negative data.
•
The models have the important properties of consistency and units invariance.
•
The models allow various types of returns to scale, and avoid division by zero.
•
We apply the models to the Taiwanese electric machinery industry.
•
We provide data-oriented and application-oriented extensions.

Abstract

This paper proposes slacks-based measure (SBM) data envelopment analysis (DEA) models that handle negative data. Unlike existing negative data allowable DEA models, the proposed SBM DEA models are consistent with ordinary SBM models and units invariant, they handle various types of returns to scale, and they avoid division by zero. These new SBM DEA models transform original negative inputs and outputs into positive counterparts based on a newly defined “base point”. Hence, these models are referred to as the BP-SBM DEA models. In addition to the basic BP-SBM DEA models, this research further develops data-oriented and application-oriented BP-SBM DEA-type models for different application problems involving negative data. Numerical examples are provided to illustrate various aspects and implementation details of these models.

Introduction

Since the publication of the seminal paper by Charnes, Cooper and Rhodes (1978), various types of data envelopment analysis (DEA) models have been developed; the proposed DEA approach has been used to measure performance in fields such as finance, banking, transportation, logistics, high-technology industry, and medicine (see, e.g., Tone, 2017). It is interesting to note that more sophisticated and various DEA models are constantly being developed due to the inability of existing DEA models to tackle newly confronted applications. For example, in many applications, the appropriate DEA models would be units invariant and able to handle negative data.

Broadly speaking, DEA models can be classified into radial and non-radial models, which have very different characteristics. Radial DEA models (e.g., the CCR (Charnes–Cooper–Rhodes) model) deal with proportional changes of inputs or outputs, whereas non-radial DEA models (e.g., the slacks-based measure (SBM) model) directly handle input- or output-slacks without assuming proportional changes of inputs or outputs. This research focuses on non-radial DEA models and intends to develop various new SBM-type DEA models. The main goal is to broaden the application fields of the SBM DEA approach by empowering it to encompass more comprehensive characteristics that are needed to conduct an effective performance evaluation in a variety of application areas.

The original SBM DEA model, proposed in Tone (2001), satisfies the following two properties:

(1)
Units-invariant: the measure should be invariant with respect to the units of data.
(2)
Monotonic: the measure should be monotone decreasing in each slack in input and output.

However, this SBM DEA model cannot deal with negative data, which are encountered in many applications, such as financial portfolio optimization in which some assets may naturally have negative expected returns. This shortcoming significantly limits the wide applications of the original SBM DEA model. That is, sophisticated, applicable SBM-type DEA models entail the following crucial properties for effectively measuring the efficiency of Decision Making Units (DMUs).

(a)
Accepts negative data,
(b)
Ensures units invariance,
(c)
Prevents division by zero irrationality, and
(d)
Maintains consistency with ordinary SBM models.

Therefore, this research aims to develop various SBM DEA models, referred to as the BP-SBM DEA models, which are not only units invariant, but also handle negative data. There are a variety of application areas involving negative data. Hence, in addition to the basic BP-SBM DEA models, this research further develops data-oriented (i.e., concerning specific data structures) and application-oriented (e.g., weighted, super-efficiency and SBM_Max) BP-SBM DEA-type models for application areas like these that involve negative data.

However, we note that even though the proposed BP-SBM DEA models are not translation invariant, this is also a desirable property for DEA models. So far, little work has been done to address translation invariance in DEA with negative data. Although Ali and Seiford (1990) proposed the first research on translation invariance in DEA, they consider only non-negative data. Lovell and Pastor (1995) observe that variable returns to scale (VRS) is a must for DEA models to ensure that translation invariance holds, but VRS DEA models may not be fully translation invariant. For example, BCC models (Banker, Charnes & Cooper, 1984) are invariant with respect to translation of inputs or outputs, but not both. Furthermore, to the best of our knowledge, Sharp, Meng and Liu (2007) present, so far, the only SBM DEA model that addresses the translation invariance issue with negative data. However, their model is confined to the VRS assumption. Since this research also focuses on SBM-type DEA models, we will later contrast the models in Sharp et al. (2007) with ours, and refer the reader to Pastor and Aparicio (2015) for an overview of the different approaches that have considered translation invariant DEA models. Moreover, it is important to note that as mentioned in Pastor and Aparicio (2015), after displacing negative data with transformed non-negative data in the BCC model, its corresponding efficient frontier remains unchanged, but its corresponding efficiency score is not invariant. Actually, the fact that displacement can alter the actual efficiency score may be seen in other translation invariant DEA models. Finally, note also that there are few DEA models that are both units invariant and translation invariant (see, e.g., Cheng, Zervopoulos & Qian, 2013; Lin & Chen, 2017; Lovell & Pastor, 1995).

The remainder of this paper unfolds as follows. Section 2 presents the basic BP-SBM DEA models. Section 3 uses several illustrative examples to demonstrate these models. In Section 4, we provide a data-oriented extension. More specifically, we derive different base points based on different available data structures. Section 5 contrasts the proposed approaches with those in Sharp et al. (2007). Section 6 extends the models in Section 2 to develop various BP-SBM DEA-type models to meet different application needs. Section 7 applies the developed models to a real-life case. Section 8 concludes this paper.

Section snippets

BP-SBM DEA models

This section presents the basic BP-SBM DEA model and shows that the model possesses the properties of continuity, units invariance, and monotonicity. However, to begin with, we introduce how to set the minimum values of inputs and outputs and use these values to define the base point.

Illustrative examples

This section uses several examples to illustrate how DEA analysis is performed with the proposed BP-SBM DEA model. In addition, we examine the effect of varying σ, τ, ρ, and γ in (2) and (3) on the DEA efficiency scores, and the units invariance property of the model.

Data-oriented extensions

In this section, we introduce two extensions of the model in (5) due to specific data structures.

(a) Suppose that an output variable, say Output 1, has an upper bound u₁(>0), e.g., the capacity of a stadium or the maximum percentage of 100%. Then, letting σ be a positive number, we set $y_{1}^{\max} = u_{1} (1 + σ) .$

The distance $y_{1}^{\max} - y_{1 o} (> 0)$ indicates the closeness of DMU_o’s Output 1 to the upper bound of Output 1. Obviously, the shorter the distance, the more efficient the DMU. Thus, we move Output 1 to the

Contrast to Sharp et al. (2007)

Sharp et al. (2007) propose an SBM DEA model similar to ours. As mentioned earlier, Sharp et al. (2007) is, to our knowledge, the only SBM DEA model in the literature that deals with translation invariance with negative data. Their model uses the following scheme to accept negative data. That is, they first let $\begin{matrix} P_{r o}^{+} = max_{j} (y_{r j}) - y_{r o} r = 1, . . ., s \\ P_{i o}^{-} = x_{i o} - min_{j} (x_{i j}) i = 1, . . ., m . \end{matrix}$

We note that these measures were first proposed by Portela, Thanassoulis and Simpson (2004), who refer to them as the range of

Application-oriented extensions

Our proposed approaches for handling negative data can be applied to other types of slacks-based measure DEA models. Thus, in this section, we show how to extend the basic BP-SBM DEA models presented in Section 2 to develop other BP-SBM DEA-type models to meet different application needs.

Application to Taiwanese electrical machinery industry

The application tool for managerial evaluation should consider publicly available information. In this section, we implement the proposed BP-SBM DEA models to evaluate the 30 Taiwanese electrical machinery listed firms in 2017. All data are collected from the Taiwan Economic Journal database. Two input indicators are considered: Cost_of_sales and R&D expenses. Cost_of_sales represents the cost of goods or services supplied in a given accounting period, while R&D expenses implies the intention

Conclusions

The DEA approach has been widely applied to various areas for conducting performance evaluation. Users, however, are expressing a need for more sophisticated and applicable DEA models. Indeed, it is becoming a must for DEA models to boast such properties as units invariance and negative-data acceptableness. It is emphasized that the ability to deal with negative data is especially critical when applying the DEA approach to practical performance evaluation problems.

To date, several radial DEA

References (18)

K. Ando et al.
Monotonicity of minimum distance inefficiency measures for data envelopment analysis
European Journal of Operational Research
(2017)
J. Aparicio et al.
Closest targets and strong monotonicity on the strongly efficient frontier in DEA
Omega
(2014)
A. Charnes et al.
Measuring the efficiency of decision making units
European Journal of Operational Research
(1978)
ChengG. et al.
A variant of radial measure capable of dealing with negative inputs and outputs in data envelopment analysis
European Journal of Operational Research
(2013)
H. Fukuyama et al.
Input–output substitutability and strongly monotonic p-norm least distance DEA measure
European Journal of Operational Research
(2014)
K. Tone
A slacks-based measure of super-efficiency in data envelopment analysis
European Journal of Operational Research
(2002)
A.I. Ali et al.
Translation invariance in data envelopment analysis
Operations Research Letters
(1990)
R.D. Banker et al.
Some models for estimating technical and scale inefficiencies in data envelopment analysis
Management Science
(1984)
W.W. Cooper et al.
A range adjusted measure of inefficiency for use with additive models and relations to other models and measures in DEA
Journal of Productive Analysis
(1999)

There are more references available in the full text version of this article.

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Citation Excerpt :
As a consequence, it might misguide decision-makers while adopting strategies to improve an inefficient DMU. However, Tone et al. (2020) did not adopt any specific approach to determine the exact values of these terms. Moreover, the model is unable to satisfy the property of translation invariance.
For most of the Slacks-Based Measure (SBM)-type models in Data Envelopment Analysis, the variable returns-to-scale (VRS) condition is required to become feasible and translation invariant while applied on negative data. Unlike them, the “Base Point SBM (BP-SBM)” and “Super BP-SBM” models accept negative values under four returns-to-scale (RTS) conditions; however, (i) they are not translation invariant, (ii) they are sensitive to some perturbation terms which help to translate negative data to positive data by avoiding division-by-zero irrationality, and (iii) the Super BP-SBM model suffers from infeasibility issue. To address the above limitations, this study improves the existing BP-SBM and Super BP-SBM models. The Improved BP-SBM (IBP-SBM) and Improved Super BP-SBM (ISBP-SBM) models are translation invariant under four different RTS conditions, and do not involve any perturbation terms. Although the proposed ISBP-SBM model is feasible under the four RTS conditions, it fails to provide Pareto efficient projections. Therefore, we propose a two-stage super-efficiency approach, where the ISBP-SBM model is applied on the first stage to obtain the input savings and output surpluses, which are incorporated in the IBP-SBM model to be applied on the second stage. To validate the proposed models, their results are compared with some state-of-the-art models using a benchmark dataset. Experimental findings indicate that the proposed models have overcome the limitations of the existing models while applied on negative data. Further, three real-life case studies of supplier selection, financial performance evaluation, and airline performance evaluation are used to discuss the applicability of the proposed approach.

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View full text

Decision SupportHandling negative data in slacks-based measure data envelopment analysis models

Highlights

Abstract

Introduction

Section snippets

BP-SBM DEA models

Illustrative examples

Data-oriented extensions

Contrast to Sharp et al. (2007)

Application-oriented extensions

Application to Taiwanese electrical machinery industry

Conclusions

European Journal of Operational Research

Omega

European Journal of Operational Research

European Journal of Operational Research

European Journal of Operational Research

European Journal of Operational Research

Translation invariance in data envelopment analysis

Operations Research Letters

Some models for estimating technical and scale inefficiencies in data envelopment analysis

Management Science

A range adjusted measure of inefficiency for use with additive models and relations to other models and measures in DEA

Journal of Productive Analysis

Decision Support
Handling negative data in slacks-based measure data envelopment analysis models