Evaluation of decision-making units using directional distance function with weak disposability in the presence of undesirable outputs

[1] A. Alfredsson, J. Mansson, P. Vikstrom, Internalising external environmental effects in efficiency
analysis: The Swedish pulp and paper industry 2000–2007, Econ. Anal. Policy. 51 (2016) 22–31.
[2] K. Asanimoghadam, M. Salahi, A. Jamalian and R. Shakouri, A two-stage structure with undesir-
able outputs: slacks based and additive slacks-based measures DEA models, RAIRO-Oper. Res. 56
(2022) 2513-2534.
[3] R. U. Ayres, A. V. Kneese, Production, consumption, and externalities, Am. Econ. Rev. 59 (1969)
282–297.
[4] R. Ayres, Thermodynamics and process analysis for future economic scenarios, Env. Resour. Econ.
6 (1995) 207–230.
[5] S. A. Berg, F. R. Førsund, E. S. Jansen, Malmquist indices of productivity growth during the dereg-
ulation of Norwegian banking, 1980–1989, Scand. J. Econ. (1992) S211–S228.
[6] R. G. Chambers, Y. Chung, R. Fare, Benefit and distance fuctions, J. Econ. Theory. 70 (1996)
407–419.
[7] A. Charnes, W. W. Cooper, E. Rhodes, Measuring the efficiency of decision making units, Eur. J.
Oper. Res. 2 (1978) 429–444.
[8] P. C. Chen, M. M. Yu, C. C. Chang, S. H. Hsu, S. Managi, The enhanced Russell-based direc-
tional distance measure with undesirable outputs: Numerical example considering CO2 emissions,
Omega. 53 (2015) 30–40.
[9] Y. H. Chung, R. Fare, S. Grosskopf, Productivity and undesirable outputs: a directional distance
function approach, J. Environ. Manag. 51 (1997) 229–240.
[10] Q. Cui, Investigating the airlines emission reduction through carbon trading under CNG2020 strat-
egy via a Network Weak Disposability DEA, Energy. 180 (2019) 763–771.
[11] K. H. Dakpo, P. Jeanneaux, L. Latruffe, Modelling pollution-generating technologies in perfor-
mance benchmarking: Recent developments, limits and future prospects in the nonparametric
framework, Eur. J. Oper. Res. 250 (2016) 347–359.
[12] R.G. Dyson, R. Allen, A.S. Camanho, V.V. Podinovski, C.S. Sarrico, E.A. Shale, Pitfalls and
protocols in DEA, Eur. J. Oper. Res. 132 (2001) 245–259.
[13] Enerdata, World Energy & Climate Statistics (Enerdata), Available at:
https://yearbook.enerdata.net/, Accessed: 11 May 2022.
[14] A.R. Fakharzadeh Jahromi, H. Rostamzadeh, M.H. Mojtabaei, Designing and evaluating an op-
timal budget plan for parallel network systems through DEA methodology, J. Math. Model. 11
(2023) 665–680.

[15] R. F¨are, S. Grosskopf, Nonparametric productivity analysis with undesirable outputs: comment,
Am. J. Agric. Econ. 85 (2003) 1070–1074.
[16] R. F¨are, S. Grosskopf, Modeling undesirable factors in efficiency evaluation: comment, Eur. J.
Oper. Res. 157 (2004) 242–245.
[17] R. Fare, S. Grosskopf, Directional distance functions and slacks-based measures of efficiency, Eur.
J. Oper. Res. 200 (2010) 320–322.
[18] K. He, L. Jie, Evaluation of the eco-efficiency of the industrial, system in China considering re-
gional heterogeneity and dynamics, Oper. Res. 25 (2025) 1–36.
[19] International Renewable Energy Agency (IRENA), RENEWABLE CAPACITY STATISTICS 2021,
Available at: www.irena.org.
[20] C. Kao, Network data envelopment analysis, Int. Ser. Oper. Res. Manag. Sci., 2017,
https://doi.org/10.1007/978-3-319-31718-2.
[21] C. Kao, S.N. Hwang, Measuring the effects of undesirable outputs on the efficiency of production
units, Eur. J. Oper. Res. 292 (2021) 996–1003.
[22] C. Kao, S.N. Hwang, Separating the effect of undesirable outputs generation from the inefficiency of
desirable outputs production in efficiency measurement, Eur. J. Oper. Res. 311 (2023) 1097–1102.
[23] G. Karagiannis, S. Kourtzidis, On modelling non-performing loans in bank efficiency analysis, Int.
J. Financ. Econ. (2024).
[24] M. Karami Khorramabadi, M. Yarahmadi, M. Ghiyasi, Cost Efficiency Evaluation Based on a Data
Envelopment Analysis Approach by Considering Undesirable Outputs on the Basis of the Semi-
Disposability Assumption, Math. Comput. Appl. 25 (2020) 4.
[25] R. Kazemi Matin, A review on weak disposability axiom in modern undesirable output in nonpara-
metric activity analysis, Oper. Res. Appl. 231 (2011) 53–69.
[26] T.C. Koopmans, An analysis of production as an efficient combination of activities, Act. Anal. Prod.
Alloc. (1951).
[27] T. Kuosmanen, Weak disposability in nonparametric production analysis with undesirable outputs,
Am. J. Agric. Econ. 87 (2005)1077–1082.
[28] T. Kuosmanen, R. Kazemi Matin, Duality of weakly disposable technology, Omega. 39 (2011)
54–512.
[29] T. Kuosmanen, V. Podinovski Weak disposability in nonparametric production analysis: reply to
Fare and Grosskopf, Am. J. Agric. Econ. 91 (2009) 539–545.
[30] B.L. Lee, C. Wilson, C.A. Pasurka Jr., H. Fujii, S. Managi, Sources of airline productivity from
carbon emissions: An analysis of operational performance under good and bad outputs, J. Prod.
Anal. 47 (2017) 223–246.

[31] X. Lei, R. Qiu, X. Chen, T. Baleˇzentis, Z.Y. Shen, Evaluating the impact of carbon emission trading
on urban ecological efficiency in an emerging economy, J. Clean. Prod. 486 (2025) 144431.
[32] H. Leleu, Shadow pricing of undesirable outputs in nonparametric analysis, Eur. J. Oper. Res. 231
(2013) 474–480.
[33] Z. Ma, J. Yin, L. Yang, Y. Li, L. Zhang, H. Lv, Using Shannon Entropy to Improve the Identification
of MP-SBM Models with Undesirable Output, Entropy 24 (2022) 1608.
[34] M. Maghbouli, A. P.ourhabib Yekta, J. Jablonsky, Weak disposability in the non-parametric analy-
sis of efficiency: An interval abatement factor, oper. Res. Decis. 211 (2024) 193–203.
[35] OECD, About the OECD,Available at: https://www.oecd.org/about/, Accessed: 11 May 2022.
[36] V.V. Podinovski, T. Kuosmanen, Modelling weak disposability in data envelopment analysis under
relaxed convexity assumptions, Eur. J. Oper. Res. 211 (2011) 577–585.
[37] J. Pourmahmoud, S. Arami, Evaluating cost efficiency of decision-making units in an uncertain
environment, J. Math. Model. 12 (2024) 565–582.
[38] M. Salahi, A. Jamalian, R. Shakouri, K. Asanimoghadam, Additive Slack-Based Measure for a Two-
Stage Structure with Shared Inputs and Undesirable Feedback, Adv. Oper. Res. 1 (2022) 7596736.
[39] L.M. Seiford, J. Zhu, Modeling undesirable factors in efficiency evaluation, Eur. J. Oper. Res. 142
(2002) 16–20.
[40] L.M. Seiford , J. Zhu, A response to comments on modeling undesirable factors in efficiency eval-
uation, Eur. J. Oper. Res. 161 (2005) 579–581.
[41] R. Shakouri, M. Salahi, Stochastic P-Robust Approach to a Centralized Two-Stage DEA System
with Resource Waste, Adv. Oper. Res. 1 (2022) 1133882.
[42] R.W. Shephard, Theory of Cost and Production Functions, Princeton University Press, Princeton,
1970.
[43] S. Soofizadeh, R. Fallahnejad, A bargaining game model for performance evaluation in network
DEA considering shared inputs in the presence of undesirable outputs, J. Math. Model. 10 (2022)
227–245.
[44] T. Tamaki, H. Nakamura, H. Fujii, S. Managi, Efficiency and emissions from urban transport:
Application to world city-level public transportation, Econ. Anal. Policy. 61 (2016) 55–63.
[45] The World Bank, World Bank Open Data, Available at: https://data.worldbank.org/, Accessed: 11
May 2022.
[46] C.N. Wang, T.T. Thanh Nguyen, T.T., Dang, H.P. Hsu, Exploring economic and environmental effi-
ciency in renewable energy utilization: a case study in the Organization for Economic Cooperation
and Development countries, Environ. Sci. Pollut. Res. 30 (2023) 72949–72965.