Returns to scale in dynamic DEA

doi:10.1016/j.ejor.2003.08.055

European Journal of Operational Research

Volume 161, Issue 2, 1 March 2005, Pages 536-544

https://doi.org/10.1016/j.ejor.2003.08.055 Get rights and content

Abstract

Two different types of inputs (variable inputs and quasi-fixed inputs) are incorporated into an analytical framework of dynamic data envelopment analysis (DEA). A unique feature of the quasi-inputs is that those are considered as outputs at the current period, while being treated as inputs at the next period. The dynamic DEA can measure interdependency among consecutive periods. This study incorporates the concept of returns to scale into the dynamic DEA.

Introduction

Data envelopment analysis (DEA), first proposed by Charnes et al. (1978), is a managerial approach that has been widely applied to performance analysis in public and private sectors. Such previous DEA contributions can be found in the bibliography prepared by Tavares (2002) that contains more than 3000 research efforts in the past two decades. In addition to the DEA practicality and applicability, it is important to note that DEA is now widely disseminated from the United States to other industrial nations such as Japan and Taiwan/China (see, for example, a Japanese DEA book (Sueyoshi, 2001) and a Chinese DEA book (Kao et al., 2003)).

In the previous DEA research efforts, Nemoto and Goto (1999) added an important perspective on dynamic DEA. Their research incorporates two different types of inputs (variable inputs and quasi-fixed inputs) into a framework of dynamic DEA. A unique feature of the quasi-inputs is that those are considered as outputs at the current period, while being treated as inputs at the next period. For example, a power generator uses workers and fuels (as variable inputs) in order to produce electricity (as an output). Most of generated power is sold to wholesalers in a power market. However, a part (as a quasi-fixed output) of the generated power is internally saved within the generator. The saved power is used to generate electricity in the next period. So, it functions as a quasi-fixed input. The quasi-fixed output in the current period is internally purchased by the generator as a quasi-fixed input for the next period. Such a unique feature provides DEA with an opportunity to measure interdependency between two consecutive periods (so, eventually over an entire observed period). Consequently, a new type of DEA efficiency measure is proposed within a framework of dynamic DEA.

Acknowledging their contribution on the dynamic DEA; unfortunately, this study finds that their research has not yet documented how to measure returns to scale (RTS) in the analytical framework of the dynamic DEA. (see Sengupta (1995) for his description on another type of dynamic DEA.) Hence, we extend their approach in a manner that the concept of RTS is incorporated into the DEA dynamics.

The remaining structure of this article is organized as follows: Section 2 examines mathematical properties related to the dynamic DEA. Section 3 discusses how to measure RTS in the framework of the dynamic DEA. A concluding comment summarizes this research in Section 4.

Section snippets

Mathematical properties of dynamic DEA

To incorporate the concept of RTS into the dynamic DEA, this article starts with a description on the approach proposed by Nemoto and Goto (NG, 1999). It is assumed that there are N decision making units (DMUs; i=1,2,…,N) and their production activities are examined in T periods (t=1,2,…,T). In the tth period, each DMU_i uses two different groups of inputs: k_t−1,i (an ℓ-dimensional vector of quasi-fixed inputs) and x_t,i (an m-dimensional vector of variable inputs) in order to produce two

Measurement of RTS

A major difficulty related to the RTS measurement is that a supporting hyperplane of Ψ_t cannot be uniquely determined. In other words, all the dual variables, including σ_t, have often multiple solutions. NG (1999) have discussed all their argument under the assumption that the adjustment cost $θ_{t−1}^{∗}$ is always unique.)

Conclusion and future extensions

In this study, two different types of inputs (variable inputs and quasi-fixed inputs) are incorporated into dynamic DEA. The quasi-inputs are considered as outputs at the current period, while being treated as inputs at the next period. Such a unique feature provides DEA with an opportunity to investigate interdependency among consecutive periods. This study extends the approach in a manner that the concept of RTS is incorporated into the DEA dynamics. The RTS measurement proposed in this study

Acknowledgements

The authors thank Dr. Wallenius and two reviewers for their constructive comments which improve the quality of this article. This research was prepared when Dr. Sekitani was a visiting professor at New Mexico Tech. The second author acknowledges the support of the Ministry of Education, Science, Sports and Culture of Japan, Grant-in-Aid for Scientific Research(C), 15510123.

References (8)

A. Charnes et al.
Measuring the efficiency of decision making units
European Journal of Operational Research
(1978)
J. Nemoto et al.
Dynamic data envelopment analysis: modeling intertemporal behavior of a firm in the presence of productive inefficiencies
Economics Letters
(1999)
C. Kao et al.
Performance Evaluation: Data Envelopment Analysis (in Chinese)
(2003)
J.k. Sengupta
Dynamics of Data Envelopment Analysis, Theory of Systems Efficiency
(1995)

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

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Interfaces with Other DisciplinesReturns to scale in dynamic DEA

Abstract

Introduction

Section snippets

Mathematical properties of dynamic DEA

Measurement of RTS

Conclusion and future extensions

Acknowledgements

European Journal of Operational Research

Economics Letters

Performance Evaluation: Data Envelopment Analysis (in Chinese)

Dynamics of Data Envelopment Analysis, Theory of Systems Efficiency

Interfaces with Other Disciplines
Returns to scale in dynamic DEA