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

Document Type : Research Article

Authors

Department of Applied Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran

Abstract

This study evaluates decision-making units (DMUs) that produce  unavoidable undesirable outputs,  which negatively affect performance. Previous studies by researchers have explored   approaches for calculating the minimum undesirable outputs, presenting the abatement range for undesirable outputs and assessing DMU efficiency, yet challenges remain.  To address this, we  introduce  a new method based on the directional distance function  model with individual-proportion weak disposability. The proposed method calculates the minimum unavoidable undesirable outputs, evaluates DMU  efficiency  for both undesirable and minimum undesirable outputs, separates the effects  of desirable and undesirable outputs on inefficiency, and resolves the limitations of earlier methods. Finally, the proposed method is applied to practical examples to demonstrate its superiority over existing approaches.  

Keywords

Main Subjects


[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.