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Identifying the Global Reference Set in DEA: An Application to the Determination of Returns to Scale

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Handbook of Operations Analytics Using Data Envelopment Analysis

Part of the book series: International Series in Operations Research & Management Science ((ISOR,volume 239))

Abstract

In data envelopment analysis (DEA), any set of observed decision making units (DMUs) that produce a projection point of an inefficient DMU is called a reference set of this DMU. Based on this definition, however, the concept of reference set is not mathematically well defined in the non-radial DEA setting. This is because a given projection point may be generated by multiple unary reference sets, and different projection points may result in multiple maximal reference sets. In this chapter, first, we address this issue by differentiating between the uniquely found reference set, called the global reference set (GRS), and the unary and maximal types of the reference set for which the multiplicity issue may occur. Second, to identify the GRS, we propose a general linear programming based approach that is computationally more efficient than its alternatives. Third, we define the returns to scale (RTS) of an inefficient DMU at its projection point that is produced by all—but not some—of the units in its GRS. By this definition, the notion of RTS is unambiguous, since the GRS is unique and the projection points generated by all the possible reference units all exhibit the same type of RTS. Fourth, using a non-radial DEA model, we develop two precise multiplier- and envelopment-based methods to determine RTS possibilities of the DMUs. To demonstrate the ready applicability of our approach, we finally conduct an empirical analysis based on a real-life data set.

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Notes

  1. 1.

    For more details about these conditions, the interested readers may refer to Mehdiloozad et al. (2016).

  2. 2.

    The simplex algorithm for bounded variables was published by Dantzig (1955) and was independently developed by Charnes and Lemke (1954). This algorithm is much more efficient than the ordinary simplex algorithm for solving the LP problem with upper-bounded variables (Winston 2003).

  3. 3.

    See, e.g., Banker et al. (1996), Førsund (1996), Sahoo et al. (1999), Fukuyama (2000, 2001, 2003), Tone and Sahoo (2003, 2004, 2005), Sengupta and Sahoo (2006), Sahoo (2008), Podinovski et al. (2009), Podinovski and Førsund (2010), Sahoo et al. (2012), Sahoo and Tone (2013, 2015), Sahoo and Sengupta (2014), Sahoo et al. (2014a, b), among others.

  4. 4.

    Under such an occurrence, the determination of RTS via Tone’s (1996, 2005) method may be problematic. For a detailed discussion on this issue, interested readers may refer to the illustrative Figs. 1 and 2 in Krivonozhko et al. (2012c).

  5. 5.

    The occurrence of multiple MRSs is illustrated via an example in Sect. 3 in Sueyoshi and Sekitani (2007b).

  6. 6.

    For a graphical illustration of the minimum face, see Fig. 4 in Sueyoshi and Sekitani (2007b).

  7. 7.

    To the best of our knowledge, the other studies on identification of all the possible reference DMUs using the BCC model include Sueyoshi and Sekitani (2007a), Jahanshahloo et al. (2008), Krivonozhko et al. (2012a), and Roshdi et al. (2014).

  8. 8.

    For more details about the facial structure of the CRS- and VRS-based technologies, see, e.g., Davtalab-Olyaie et al. (2014a; b) and Jahanshahloo et al. (2013).

  9. 9.

    While dealing with the estimation of a piecewise log-linear technology , one may encounter negative data since the log transformation of values less than 1 are always negative (Mehdiloozad et al. 2014). One may also refer to, e.g., Pastor and Ruiz (2007), Sahoo and Tone (2009) and Sahoo et al. (2012), among others, for several examples of applications with negative data.

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Authors and Affiliations

  1. Department of Mathematics, College of Sciences, Shiraz University, Shiraz, 71454, Iran

    Mahmood Mehdiloozad

  2. Xavier Institute of Management, Xavier University, Bhubaneswar, 751013, India

    Biresh K. Sahoo

Authors
  1. Mahmood Mehdiloozad
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  2. Biresh K. Sahoo
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Correspondence to Biresh K. Sahoo .

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Editors and Affiliations

  1. Dept of Business Administration, Ming Chuan University, Taipei, Taiwan

    Shiuh-Nan Hwang

  2. College of Maritime Science & Mgmt, National Taiwan Ocean University, Keelung, Taiwan

    Hsuan-Shih Lee

  3. School of Business, Worcester Polytechnic Institute, Worcester, Massachusetts, USA

    Joe Zhu

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Mehdiloozad, M., Sahoo, B.K. (2016). Identifying the Global Reference Set in DEA: An Application to the Determination of Returns to Scale. In: Hwang, SN., Lee, HS., Zhu, J. (eds) Handbook of Operations Analytics Using Data Envelopment Analysis. International Series in Operations Research & Management Science, vol 239. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-7705-2_12

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