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Average lexicographic efficiency decomposition in two-stage data envelopment analysis: an application to China’s regional high-tech innovation systems

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Abstract

This paper proposes a lexicographic decomposition strategy for the two-stage network DEA model. Instead of assigning a priority over the two component stages, our proposed approach employs a lexicographic algorithm in such a way that the system efficiencies of the DMUs are lexicographically computed for each of their possible sequences while maintaining the efficiencies of those units already investigated unchanged. In particular, the system efficiency and the two component efficiencies can be uniquely determined under our proposed decomposition approach. Besides, prior to implement the proposed lexicographic evaluation procedure, we also highlight a potential infeasibility problem arising from the normalization constraint, and develop a modified iterative approach that facilitates us to guarantee a feasible search procedure. Finally, we use a numerical example to illustrate the effectiveness of the proposed approach, and conduct an empirical study to analyze China’s regional high-tech innovation systems for the period of 12th Five-Year Plan (2011–2015).

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Notes

  1. In order to guarantee the uniqueness of the multipliers used for calculating the component efficiencies, Kao and Liu (2019) maximized the aggregate efficiency of all DMUs for the arbitrary most important division while maintaining the aggregate system efficiency of the other DMUs unchanged. However, such a practice cannot guarantee unique efficiency decomposition.

  2. Herein, it is worth noting that there are generally two strategies to decompose the system efficiency under the two-stage production structure, namely the multiplicative decomposition strategy (MDS) proposed by Kao and Hwang (2008) and the additive decomposition strategy (ADS) proposed by Chen et al. (2009). For the MDS, it interprets the system efficiency as the product of two sub-stage efficiencies, i.e., \(\theta_{o} = \theta_{o}^{1} \times \theta_{o}^{2}\). However, this approach suffer from several shortcomings, for example, it fails to handle the situations in which a VRS assumption is required, and it becomes nonlinear when additional constraints are included. To resolve these problems, Chen et al. (2009) aggregate the components of a two-stage process by interpreting the system efficiency as the weighted average of component efficiencies. In doing so, as we will introduce below, the corresponding model can be easily transformed into a linear one.

  3. In doing so, we can always guarantee a bounded solution to model (12). Otherwise, model (12) may be unbounded as the objective of this model is to maximize the value of \(\sum\nolimits_{i = 1}^{m} {\varpi_{i} x_{{i\sigma \left( {\ell + 1} \right)}} } + \sum\nolimits_{d = 1}^{D} {\zeta_{d} z_{{d\sigma \left( {\ell + 1} \right)}} }\), which may be infinity as one only restricts that the overall efficiency of concerned DMUs should be maintained.

  4. We do not include Qinghai, Tibet, Hong Kong, Macao, and Taiwan due to data limits.

  5. Although not all inventions can be regarded as a patent, and the patents are actually differing greatly in quality (Chen, et al., 2018), the PA is indeed a reasonable and reliable indicator of unobserved innovation output.

  6. Specifically, to guarantee the feasibility of our concerned models, according to Proposition 1, we set \(\varepsilon = 10^{ - 8}\). In addition, following Lee et al. (2019), our study focuses primarily on the VRS condition.

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Acknowledgements

The authors would especially like to thank the editors, and three anonymous reviewers whose comments led to significant improvements in the paper. This work is supported by the National Natural Science Foundation of China under grant (No. 71671095), the Tianjin Philosophy and Social Science Planning Key Project (TJGL21-009), and the Fundamental Research Funds for the Central Universities (No. 2232021E-10).

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Authors and Affiliations

  1. Glorious Sun School of Business and Management, Donghua University, No. 1882 West Yan’an Road, Shanghai, 200051, China

    Jiawei Yang

  2. Department of Management Science and Engineering, Business school, Nankai University, Nankai District, No. 121 Baidi Road, Tianjin, 300071, China

    Lei Fang

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Appendix

Appendix

See Tables 13, 14, 15, 16, 17.

Table 13 The system efficiency of 29 provincial-level regions using the traditional ADA
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Table 14 The R&D efficiency of 29 provincial-level regions using the traditional ADA (under strategy I)
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Table 15 The commercialization efficiency of 29 provincial-level regions using the traditional ADA (under strategy I)
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Table 16 The R&D efficiency of 29 provincial-level regions using the traditional ADA (under strategy II)
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Table 17 The commercialization efficiency of 29 provincial-level regions using the traditional ADA (under strategy II)
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Yang, J., Fang, L. Average lexicographic efficiency decomposition in two-stage data envelopment analysis: an application to China’s regional high-tech innovation systems. Ann Oper Res 312, 1051–1093 (2022). https://doi.org/10.1007/s10479-021-04427-z

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