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Output, input, and undesirable output interconnections in data envelopment analysis: convexity and returns-to-scale

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Abstract

Increasing attention has been given to the development of specific techniques to deal with interconnections between outputs, inputs, and undesirable outputs for data envelopment analysis (DEA) models. These techniques offer the advantages of improving the realism and the flexibility of DEA models; two aspects of crucial importance to convince practitioners about the attractiveness and the reliability of DEA models. In this paper, we propose a unifying methodology coherent with previous works to model these interconnections. We suggest treating the outputs as the fundamental component of the production process by modelling every output individually. This gives us the option of considering the interconnections with the inputs and the undesirable outputs. In particular, we make a distinction between undesirable outputs/inputs that are due to/used by all the outputs, and those that are due/allocated to specific outputs. Attractively, our methodology also offers the option of setting a different returns-to-scale assumption for each output-specific production process, and to choose between different types of convexity. We demonstrate the usefulness of our methodology with the case of the US electricity plants producing fossil and non-fossil electricity generation.

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Notes

  1. See, for example, Färe et al. (1994), Cooper et al. (2004, 2007), Fried et al. (2008), and Cook and Seiford (2009) for reviews.

  2. See Färe and Primont (1995) for more discussion about regularity conditions of DEA models.

  3. Recent contributions on DEA models with undesirable outputs could be found in Liu et al. (2010, 2015), Chen (2014), Maghbouli et al. (2014), Bi et al. (2015), Cherchye et al. (2015), and Izadikhah and Saen (2018); for reviews, refer, for example, to Zhou et al. (2008) and Dakpo et al. (2016).

  4. See, for example, Podinovski (2004b, c, 2018), Tone and Sahoo (2006), Lozano and Villa (2010), Tone (2011), Alirezaee et al. (2018), and Perez-Lopez et al. (2018) for DEA models with returns-to-scale; and Banker et al. (2004), Banker et al. (2011), and Sahoo and Tone (2015) for reviews.

  5. See Podinovski (2009), Podinovski et al. (2014, 2018), Afsharian et al. (2015), and Podinovski and Husain (2017) for extensions.

  6. At this point, we remark that Afriat (1972) was the first to introduce efficiency analysis without the assumption of convexity for single output case (and using different terminology).

  7. See, for example, Bogetoft et al. (2000), Dekker and Post (2001), Kuosmanen (2001, 2003), Briec et al. (2004), Agrell et al. (2005), Podinovski (2005), Ehrgott and Tind (2009), and Podinovski and Kuosmanen (2011) for DEA models without and with partial convexity.

  8. Note that even when weak disposability is assumed, opting for a relaxed convexity approach is also of interest. See Podinovski and Kuosmanen (2011).

  9. For example, radial efficiency measurements, non-radial efficiency measurements, hybrid efficiency measurements, directional distance functions, etc. All these efficiency measurements can be considered here.

  10. See, for example, Cherchye et al. (2015) for an empirical study of electricity plants when specific targets are specified for the greenhouse gas reductions. These types of targets could fairly easily be integrated into our model.

  11. We remark that there is a debate in the literature about the proper way to evaluate the effect of exogenous factors on efficiency scores. Using a truncated regression seems to be a good compromise. See, for example, Hoff (2007) and McDonald (2009) for more discussion.

References

  • Abbott, M. (2006). The productivity and efficiency of the Australian electricity supply industry. Energy Economics, 28, 444–454.

    Google Scholar 

  • Afriat, S. (1972). Efficiency estimation of production functions. International Economic Review, 13, 568–598.

    Google Scholar 

  • Afsharian, M., Ahn, H., & Alirezaee, M. (2015). Developing selective proportionality on the FDH models: New insight on the proportionality axiom. International Journal of Information and Decision Sciences, 7, 99–114.

    Google Scholar 

  • Agrell, P. J., Bogetoft, P., Brock, M., & Tind, J. (2005). Efficiency evaluation with convex pairs. Advanced Modeling and Optimization, 7, 211–237.

    Google Scholar 

  • Alirezaee, M., Hajinezhad, E., & Paradi, J. C. (2018). Objective identification of technological returns to scale for data envelopment analysis models. European Journal of Operational Research, 266, 678–688.

    Google Scholar 

  • Banker, R. D., Charnes, A., & Cooper, W. W. (1984). Some models for estimating technical and scale efficiencies in data envelopment analysis. Management Science, 30, 1078–1092.

    Google Scholar 

  • Banker, R. D., Cooper, W. W., Seiford, L. M., Thrall, R. M., & Zhu, J. (2004). Returns to scale in different DEA models. European Journal of Operational Research, 154, 345–362.

    Google Scholar 

  • Banker, R. D., Cooper, W. W., Seiford, L. M., & Zhu, J. (2011). Returns to Scale in DEA. In W. Cooper, L. Seiford, & J. Zhu (Eds.), Handbook on data envelopment analysis. International series in operations research & management science (Vol. 164). Boston, MA: Springer.

    Google Scholar 

  • Bi, G., Luo, Y., Ding, J., & Liang, L. (2015). Environmental performance analysis of Chinese industry from a slacks-based perspective. Annals Of Operations Research, 228, 65–80.

    Google Scholar 

  • Bogetoft, P. (1996). DEA on relaxed convexity assumptions. Management Science, 42, 457–465.

    Google Scholar 

  • Bogetoft, P., Tama, J. M., & Tind, J. (2000). Convex input and output projections of nonconvex production possibility sets. Management Science, 46, 858–869.

    Google Scholar 

  • Briec, W., Kerstens, K., & Vanden, Eeckaut P. (2004). Non-convex technologies and cost functions: Definitions, duality and nonparametric tests of convexity. Journal of Economics, 81, 155–192.

    Google Scholar 

  • Charnes, A., Cooper, W. W., & Rhodes, E. (1978). Measuring the efficiency of decision making units. European Journal of Operational Research, 2, 429–444.

    Google Scholar 

  • Chen, C.-M. (2014). Evaluating eco-efficiency with data envelopment analysis: An analytical reexamination. Annals of Operations Research, 214, 49–71.

    Google Scholar 

  • Cherchye, L., De Rock, B., Dierynck, B., Roodhooft, F., & Sabbe, J. (2013). Opening the black box of efficiency measurement: Input allocation in multi-output settings. Operations Research, 61, 1148–1165.

    Google Scholar 

  • Cherchye, L., De Rock, B., & Walheer, B. (2015). Multi-output efficiency with good and bad outputs. European Journal of Operational Research, 240, 872–881.

    Google Scholar 

  • Cherchye, L., De Rock, B., & Walheer, B. (2016). Multi-output profit efficiency and directional distance function. OMEGA, 61, 100–109.

    Google Scholar 

  • Chung, Y., Färe, R., & Grosskopf, S. (1997). Productivity and undesirable outputs: A directional distance function approach. Journal of Environmental Management, 51, 229–240.

    Google Scholar 

  • Cook, W. D., Du, J., & Zhu, J. (2017). Units invariant DEA when weight restrictions are present: Ecological performance of US electricity industry. Annals of Operations Research, 255, 323–346.

    Google Scholar 

  • Cook, W. D., & Seiford, L. M. (2009). Data envelopment analysis (DEA)-thirty years on. European Journal of Operational Research, 192, 1–17.

    Google Scholar 

  • Cooper, W. W., Seiford, L. M., & Tone, K. (2007). Data envelopment analysis: A comprehensive text with models, applications, references and DEA-solver software (2nd ed.). New York: Springer.

    Google Scholar 

  • Cooper, W. W., Seiford, L. M., & Zhu, J. (2004). Handbook on data envelopment analysis (Vol. 2). New York: Springer.

    Google Scholar 

  • Dakpo, K. H., Jeanneaux, P., & Latruffe, L. (2016). Modelling pollution-generating technologies in performance benchmarking: Recent developments, limits and future prospects in the nonparametric framework. European Journal of Operational Research, 250, 347–359.

    Google Scholar 

  • Debreu, G. (1951). The coefficient of resource utilization. Econometrica, 19(3), 273–292.

    Google Scholar 

  • Dekker, D., & Post, T. (2001). A quasi-concave DEA model with an application for branch performance evaluation. European Journal of Operational Research, 132, 296–311.

    Google Scholar 

  • Despic, O., Despic, M., & Paradi, J. (2007). DEA-R: ratio-based comparative efficiency model, its mathematical relation to DEA and its use in applications. Journal of Productivity Analysis, 28, 33–44.

    Google Scholar 

  • Ding, J., Dong, W., Liang, L., & Zhu, J. (2017). Goal congruence analysis in multi-Division Organizations with shared resources based on data envelopment analysis. European Journal of Operational Research, 263, 961–973.

    Google Scholar 

  • Du, J., Cook, W. D., Liang, L., & Zhu, J. (2014). Fixed cost and resource allocation based on DEA cross-efficiency. European Journal of Operational Research, 235, 206–214.

    Google Scholar 

  • Ehrgott, M., & Tind, J. (2009). Column generation with free replicability in DEA. Omega, 37, 943–950.

    Google Scholar 

  • Färe, R., & Grosskopf, S. (2000). Network DEA. Socio-Economic Planning Sciences, 34, 35–49.

    Google Scholar 

  • Färe, R., & Grosskopf, S. (2003). Nonparametric productivity analysis with undesirable outputs: Comment. American Journal of Agricultural Economics, 85, 1070–1074.

    Google Scholar 

  • Färe, R., & Grosskopf, S. (2004). Modeling undesirable factors in efficiency evaluation: Comment. European Journal of Operational Research, 157, 242–245.

    Google Scholar 

  • Färe, R., & Grosskopf, S. (2009). A comment on weak disposability in nonparametric production analysis. American Journal of Agricultural Economics, 91, 535–538.

    Google Scholar 

  • Färe, R., Grosskopf, S., & Lovell, C. A. K. (1994). Production frontier. Cambridge: Cambridge University Press.

    Google Scholar 

  • Färe, R., Grosskopf, S., Lovell, C. A. K., & Pasurka, C. (1989). Multilateral productivity comparisons when some outputs are undesirable: A nonparametric approach. The Review of Economics and Statistics, 71, 90–98.

    Google Scholar 

  • Färe, R., Grosskopf, S., & Whittaker, G. (2007). Network DEA. In J. Zhu & W. Cook (Eds.), Modeling data irregularities and structural complexities in data envelopment analysis. New York: Springer.

    Google Scholar 

  • Färe, R., & Primont, D. (1995). Multi-output production and duality: theory and applications. Boston: Kluwer.

    Google Scholar 

  • Farrell, M. (1957). The measurement of productive efficiency. Journal of the Royal Statistical Society Series 1, General, 120(Part, 3), 253–281.

    Google Scholar 

  • Forsund, F. R. (2018). Multi-equation modelling of desirable and undesirable outputs satisfying the materials balance. Empirical Economics, 54, 67–69.

    Google Scholar 

  • Fried, H., Lovell, C. A. K., & Schmidt, S. (2008). The measurement of productive efficiency and productivity change. Oxford: Oxford University Press.

    Google Scholar 

  • Giannakis, D., Jamasb, T., & Pollitt, M. (2005). Benchmarking and incentive regulation of quality of service: An application to the UK electricity distribution networks. Energy Policy, 33, 2256–2271.

    Google Scholar 

  • Hailu, A. (2003). Nonparametric productivity analysis with undesirable outputs: Reply. American Journal of Agricultural Economics, 85, 1075–1077.

    Google Scholar 

  • Hailu, A., & Veeman, T. S. (2001). Non-parametric productivity analysis with undesirable outputs: An application to the Canadian pulp and paper industry. American Journal of Agricultural Economics, 83, 605–616.

    Google Scholar 

  • Hoff, A. (2007). Second stage DEA: Comparison of approaches for modeling the DEA score. European Journal of Operational Research, 181, 425–435.

    Google Scholar 

  • Izadikhah, M., & Saen, R. F. (2018, forthcoming). Assessing sustainability of supply chains by chance-constrained two-stage DEA model in the presence of undesirable factors. Computers & Operations Research. https://doi.org/10.1016/j.cor.2017.10.002.

    Google Scholar 

  • Jamasb, T., & Pollitt, M. (2003). International benchmarking and regulation: An application to European electricity distribution utilities. Energy Policy, 31, 16091622.

    Google Scholar 

  • Korhonen, P. J., & Syrijanen, M. J. (2003). Evaluation of cost efficiency in finnish electricity distribution. Annals of Operations Research, 121, 105–122.

    Google Scholar 

  • Kulshreshtha, M., & Parikh, J. K. (2002). Study of efficiency and productivity growth in opencast and underground coal mining in India: A DEA analysis. Energy Economics, 24, 439–453.

    Google Scholar 

  • Kuosmanen, T. (2001). DEA with efficiency classification preserving conditional convexity. European Journal of Operational Research, 132, 326–342.

    Google Scholar 

  • Kuosmanen, T. (2003). Duality theory of non-convex technologies. European Journal of Operational Research, 20, 273–304.

    Google Scholar 

  • Kuosmanen, T. (2005). Weak disposability in nonparametric production analysis with undesirable outputs. American Journal of Agricultural Economics, 87, 1077–1082.

    Google Scholar 

  • Kuosmanen, T., & Podinovski, V. (2009). Weak disposability in nonparametric production analysis: Reply to Färe and Grosskopf. American Journal of Agricultural Economics, 91, 539–545.

    Google Scholar 

  • Li, Y. J., Yang, F., Liang, L., & Hua, Z. S. (2009). Allocating the fixed cost as a complement of other cost inputs: A DEA approach. European Journal of Operational Research, 197, 389–401.

    Google Scholar 

  • Liu, W., Meng, W., Li, X. X., & Zhang, D. Q. (2010). DEA models with undesirable inputs and outputs. Annals of Operations Research, 173, 177–194.

    Google Scholar 

  • Liu, W., Zhou, Z., Ma, C., Liu, D., & Shen, W. (2015). Two-stage DEA models with undesirable input-intermediate-outputs. Omega, 56, 74–87.

    Google Scholar 

  • Lozano, S., & Villa, G. (2010). Gradual technical and scale efficiency improvement in DEA. Annals of Operations Research, 173, 123–136.

    Google Scholar 

  • Maghbouli, M., Amirteimoori, A., & Kordrostami, S. (2014). Two-stage network structures with undesirable outputs: A DEA based approach. Measurement, 48, 109–118.

    Google Scholar 

  • McDonald, J. (2009). Using least squares and tobit in second stage DEA efficiency analyses. European Journal of Operational Research, 197, 792–798.

    Google Scholar 

  • Nehring, K., & Puppe, C. (2004). Modelling cost complementarities in terms of joint production. Journal of Economic Theory, 118, 252–264.

    Google Scholar 

  • Panzar, J. C., & Willig, R. D. (1977). Economies of scale in multi-output production. Quarterly Journal of Economics, 91, 481–493.

    Google Scholar 

  • Panzar, J. C., & Willig, R. D. (1981). Economies of scope. American Economic Review, 71(2), 268–272.

    Google Scholar 

  • Perez-Lopez, G., Prior, D., & Zafra-Gomez, J. L. (2018). Temporal scale efficiency in DEA panel data estimations. An application to the solid waste disposal service in Spain. Omega, 76, 18–27.

    Google Scholar 

  • Petersen, N. C. (1990). Data envelopment analysis on a relaxed set of assumptions. Management Science, 36, 305–314.

    Google Scholar 

  • Podinovski, V. V. (2004a). Bridging the gap between the constant and variable returns-to-scale models: Selective proportionality in data envelopment analysis. Journal of the Operational Research Society, 55, 265–276.

    Google Scholar 

  • Podinovski, V. V. (2004b). On the linearization of reference technologies for testing returns to scale in FDH models. European Journal of Operational Research, 152(3), 800–802.

    Google Scholar 

  • Podinovski, V. V. (2004c). Local and global returns to scale in performance measurement. Journal of Operational Research Society, 55(2), 170–178.

    Google Scholar 

  • Podinovski, V. V. (2005). Selective convexity in DEA models. European Journal of Operational Research, 161, 552–563.

    Google Scholar 

  • Podinovski, V. V. (2009). Production technologies based on combined proportionality assumptions. Journal of Productivity Analysis, 32, 21–26.

    Google Scholar 

  • Podinovski, V. V. (2018). Returns to scale in convex production technologies. European Journal of Operational Research, 258, 970–982.

    Google Scholar 

  • Podinovski, V. V., & Husain, W. R. W. (2017). The hybrid returns-to-scale model and its extension by production trade-offs: An application to the efficiency assessment of public universities in Malaysia. Annals of Operations Research, 250, 65–84.

    Google Scholar 

  • Podinovski, V. V., Ismail, I., Bouzdine-Chameeva, T., & Wenjuan Zhang, W. J. (2014). Combining the assumptions of variable and constant returns to scale in the efficiency evaluation of secondary schools. European Journal of Operational Research, 239, 504–513.

    Google Scholar 

  • Podinovski, V. V., & Kuosmanen, T. (2011). Modelling weak disposability in data envelopment analysis under relaxed convexity assumptions. European Journal of Operational Research, 211, 577–585.

    Google Scholar 

  • Podinovski, V. V., Olesen, O. B., & Sarrico, S. C. (2018). Nonparametric production technologies with multiple component processes. Operations Research, 66, 282–300.

    Google Scholar 

  • Pombo, C., & Taborda, R. (2006). Performance and efficiency in Colombias power distribution system: Effects of the 1994 reform. Energy Economics, 28, 339–369.

    Google Scholar 

  • Reinhard, S., Lovell, C. A. K., & Thijssen, G. J. (2002). Analysis of Environmental Variation. American Journal of Agricultural Economics, 84, 1054–1065.

    Google Scholar 

  • Sahoo, B., & Tone, K. (2015). Scale elasticity in non-parametric DEA approach. In J. Zhu (Ed.), Data envelopment analysis. International series in operations research & management science (Vol. 221). Boston, MA: Springer.

    Google Scholar 

  • Salerian, J., & Chan, C. (2005). Restricting multiple-output multiple-input DEA models by disaggregating the output-input vector. Journal of Productivity Analysis, 24, 5–29.

    Google Scholar 

  • Sarkis, J., & Cordeiro, J. J. (2012). Ecological modernization in the electrical utility industry: An application of a bads-goods DEA model of ecological and technical efficiency. European Journal of Operational Research, 219, 386–395.

    Google Scholar 

  • Scheel, H. (2001). Undesirable outputs in efficiency valuations. European Journal of Operational Research, 142, 16–20.

    Google Scholar 

  • Seiford, L. M., & Zhu, J. (2002). Modeling undesirable factors in efficiency evaluation. European Journal of Operational Research, 132, 400–410.

    Google Scholar 

  • Shepard, R. W. (1970). Theory of cost and production functions. Princeton: Princeton University Press.

    Google Scholar 

  • Silva, M. C. A. (2018). Output-specific inputs in DEA: An application to courts of justice in Portugal. Omega, 79, 43–53.

    Google Scholar 

  • Sueyoshi, T., & Goto, M. (2011). DEA approach for unified efficiency measurement: Assessment of Japanese fossil fuel power generation. Energy Economics, 33, 292–303.

    Google Scholar 

  • Sueyoshi, T., & Goto, M. (2012). DEA environmental assessment of coal fired power plants: Methodological comparison between radial and non-radial models. Energy Economics, 34, 1854–1863.

    Google Scholar 

  • Sueyoshi, T., & Goto, M. (2014). Environmental Sustainability development for supply chain management in U.S. petroleum industry by DEA environmental assessment. Energy Economics, 46, 360–374.

    Google Scholar 

  • Tone, K. (2011). On returns to scale under weight restrictions in data envelopment analysis. Journal of Productivity Analysis, 16, 31–47.

    Google Scholar 

  • Tone, K., & Sahoo, B. S. (2006). Re-examining scale elasticity in DEA. Annals of Operations Research, 145, 69–87.

    Google Scholar 

  • Tone, K., & Tsutsui, M. (2007). Decomposition of cost efficiency and its application to Japanese-US electric utility comparisons. Socio-Economic Planning Sciences, 41, 91–106.

    Google Scholar 

  • Tone, K., & Tsutsui, M. (2009). Network DEA: A slacks-based measure approach. European Journal of Operational Research, 197, 243–252.

    Google Scholar 

  • Tone, K., & Tsutsui, M. (2011). Applying an efficiency measure of desirable and undesirable outputs in DEA to U.S. electric utilities. Journal of Centrum Cathedra, 4(2), 236–249.

    Google Scholar 

  • Tulkens, H. (1993). On FDH analysis: Some methodological issues and applications to retail banking, courts, and urban transit. Journal of Productivity Analysis, 4, 183–210.

    Google Scholar 

  • Walheer, B. (2016a). A multi-sector nonparametric production-frontier analysis of the economic growth and the convergence of the European countries. Pacific Economic Review, 21(4), 498–524.

    Google Scholar 

  • Walheer, B. (2016b). Growth and convergence of the OECD countries: A multi-sector production-frontier approach. European Journal of Operational Research, 252, 665–675.

    Google Scholar 

  • Walheer, B. (2018a). Aggregation of metafrontier technology gap ratios: The case of European sectors in 1995–2015. European Journal of Operational Research, 269, 1013–1026.

    Google Scholar 

  • Walheer, B. (2018b). Cost Malmquist productivity index: An output-specific approach for group comparison. Journal of Productivity Analysis, 49(1), 79–94.

    Google Scholar 

  • Walheer, B. (2018c). Disaggregation of the Cost Malmquist Productivity Index with joint and output-specific inputs. Omega, 75, 1–12.

    Google Scholar 

  • Walheer, B. (2018d). Malmquist productivity index for multi-output producers: An application to electricity generation plants. Socio-Economic Planning Sciences. https://doi.org/10.1016/j.seps.2018.02.003.

    Google Scholar 

  • Walheer, B. (2018e). Scale efficiency for multi-output cost minimizing producers: The case of the US electricity plants. Energy Economics, 70, 26–36.

    Google Scholar 

  • Walheer, B. (2018f). Scale, congestion, and technical efficiency of European countries: A sector-based nonparametric approach. Empirical Economics. https://doi.org/10.1007/s00181-018-1426-7.

    Google Scholar 

  • Walheer, B., & Zhang, L.-J. (2018). Profit Luenberger and Malmquist–Luenberger indexes for multi-activity decision making units: The case of the star-rated hotel industry in China. Tourism Management, 69, 1–11.

    Google Scholar 

  • Yu, M.-M., Chern, C.-C., & Hsiao, B. (2013). Human resource rightsizing using centralized data envelopment analysis: Evidence from Taiwans airports. Omega, 41(1), 119–130.

    Google Scholar 

  • Zhou, P., Ang, B. W., & Poh, K. L. (2008). A survey of data envelopment analysis in energy and environmental studies. European Journal of Operational Research, 189, 1–18.

    Google Scholar 

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    Barnabé Walheer

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We thank the Editor-in-Chief Endre Boros, and the referees for their comments that have improved the paper substantially.

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Walheer, B. Output, input, and undesirable output interconnections in data envelopment analysis: convexity and returns-to-scale. Ann Oper Res 284, 447–467 (2020). https://doi.org/10.1007/s10479-018-3006-9

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