10.57647/ijm2c.2025.150427

A two-stage approach for modelling undesirable outputs in DEA with production trade-offs: A case on chain stores

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  • Javad Gerami*1,
  1. Department of Mathematics, Shiraz branch, Islamic Azad University, Shiraz, Iran.

Received: 13-07-2025

Revised: 07-09-2025

Accepted: 08-09-2025

Published in Issue 08-09-2025

Copyright (c) 2025 Javad Gerami (Author)

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

How to Cite

Gerami, J. (2025). A two-stage approach for modelling undesirable outputs in DEA with production trade-offs: A case on chain stores. International Journal of Mathematical Modelling & Computations, 15(4). https://doi.org/10.57647/ijm2c.2025.150427
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Abstract

Information about production trade-offs between inputs and outputs can be included in data envelopment analysis (DEA) models. In production processes, undesirable outputs are produced simultaneously with desirable outputs. We propose the production possibility set (PPS) with production trade-offs in the presence of undesirable outputs. This paper presents a two-stage process for measuring the efficiency of production units in the presence of undesirable outputs based on DEA with production trade-offs. In the first stage, the radial targets of the decision-making unit (DMU) under evaluation is calculated. In the second stage, we calculate the maximum amount of inefficiency slack corresponding to the components of inputs, desired outputs, and undesirable outputs. We prove that the targets obtained from the two-step process corresponding to inefficient DMUs are efficient units on the efficiency frontier of PPS. These targets have a minimum level of undesirable outputs. Also, by choosing the right direction in the presented models based on the directional distance function (DDF), we can obtain different efficient targets corresponding to each of the DMUs. We show that by changing the weight restrictions on the inputs and outputs, the efficiency score and the corresponding targets of the DMUs change. These weight restrictions are determined by the decision-maker (DM) in order to consider the importance of inputs and outputs. An application of the presented approach is presented to evaluate a set of chain stores, and at the end we present the results of the paper. ‎

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Keywords

  • Data envelopment analysis; Weight restrictions; Production trade-offs; Undesirable output; Weak disposability; Directional distance function.
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References

  1. Ali, A. I., & Seiford, L. M. (1993a). Computational accuracy and infinitesimals in data envelopment analysis. Information Systems and Operational Research, 31, 290-297. https://doi.org/10.1080/03155986.1993.11732232
  2. Ali, A. I., & Seiford, L. M. (1993b). The mathematical programming approach to efficiency analysis. In H. O. Fried, C.A.K. Lovell, &S. S. Schmidt (Eds.), The measurement of productive efficiency: Techniques and applications (pp.120–159). New York: Oxford University Press. https://doi.org/10.1093/oso/9780195072181.003.0003
  3. Allen, R., Athanassopoulos, A., Dyson, R.G., Thanassoulis, E. (1997). Weights restrictions and value judgements in data envelopment analysis: Evolution, development and future directions. Annals of Operations Research, 73, 13–34.
  4. https://doi.org/10.1023/A:1018968909638
  5. Banker, R.D., Charnes, A., Cooper, W.W. (1984). Some models for estimating technical and scale efficiencies in data envelopment analysis. Management Science, 30(9), 1078–1092.
  6. https://doi.org/10.1287/mnsc.30.9.1078
  7. Ball, V. E., Lovell, C. A. K., Nehring, R., & Somwaru, A. (1994). Incorporating undesirable outputs into models of production: An application to U.S. agriculture. Cahiers d’Économie et Sociologie Rurales, 31, 59–74.
  8. DOI: 10.3406/reae.1994.1406
  9. Charnes, A., Cooper, W.W., Rhodes, E. (1978). Measuring the efficiency of decision making units. European Journal of Operational Research, 2(6), 429–444.
  10. https://doi.org/10.1016/0377-2217(78)90138-8
  11. Chambers, R.G., Chung, Y., & Fare, R. (1996). Benefit and distance functions. Journal of Economic Theory, 70 (2), 407–419.
  12. https://doi.org/10.1006/jeth.1996.0096Get rights and content
  13. Chung, Y.H., Fare, R., and Grosskopf, S. (1997). Productivity and undesirable outputs: A directional distance function approach. Journal of Environmental Management, 51 (3), 229–240. https://doi.org/10.1006/jema.1997.0146
  14. Cook, W.D., Zhu, J. (2008). Context-dependent assurance regions in DEA. Operations Research, 56(1), 69–78. https://doi.org/10.1287/opre.1070.0500
  15. 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. https://doi.org/10.1016/j.ejor.2015.07.024
  16. Färe, R., & Grosskopf, S. (2010). Directional distance functions and slacks-based measures of efficiency. European Journal of Operational Research, 200 (1), 320–322.
  17. https://doi.org/10.1016/j.ejor.2009.01.031
  18. 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. https://www.jstor.org/stable/1928055
  19. Färe, R., Grosskopf, S., and Pasurka, C. (2013). Tradable permits and unrealized gains from trade. Energy Economics, 40, 416–424. https://doi.org/10.1016/j.eneco.2013.07.015
  20. Färe, R., Grosskopf, S., and Pasurka, C. (2014). Potential gains from trading bad out-puts: The case of U.S. electric power plants. Resource and Energy Economics, 36, 99–112.
  21. https://doi.org/10.1016/j.reseneeco.2013.11.004
  22. Kao, C. (2023). Network data envelopment analysis: Foundations and extensions. Switzerland: Springer International Publishing.
  23. https://doi.org/10.1007/978-3-031-27593-7
  24. Kao, C., & Hwang, S. N. (2021). Measuring the effects of undesirable outputs on the efficiency of production units. European Journal of Operational Research, 292, 996–1003.
  25. https://doi.org/10.1016/j.ejor.2020.11.026
  26. Kao, C., & Hwang, S. N. (2023). Separating the effect of undesirable outputs generation from the inefficiency of desirable outputs production in efficiency measurement. European Journal of Operational Research. 311, 1097–1102.
  27. https://doi.org/10.1016/j.ejor.2023.06.012
  28. Kuosmanen, T. (2005). Weak disposability in nonparametric production analysis with undesirable outputs. American Journal of Agricultural Economics, 87, 1077–1082.
  29. https://doi.org/10.1111/j.1467-8276.2005.00788.x
  30. Kuosmanen, T., & Podinovski, V. (2009). Weak disposability in nonparametric production analysis: Reply to Färe and Grosskopf. American Journal of Agricultural Economics, 91, 539–545.
  31. https://doi.org/10.1111/j.1467-8276.2008.01238.x
  32. Lin, R., & Li, Z. (2020). Directional distance based diversification super-efficiency DEA models for mutual funds. Omega, 97, 1–15. https://doi.org/10.1016/j.omega.2019.08.003
  33. Lin, R., & Liu, Q. (2021). Multiplier dynamic data envelopment analysis based on directional distance function: An application to mutual funds. European Journal of Operational Research, 293 (3), 1043–1057. https://doi.org/10.1016/j.ejor.2021.01.005
  34. Murty, S., Russell, R. R., & Levkoff, S. B. (2012). On modeling pollution-generating technologies. Journal of Environmental Economics and Management, 64, 117–135. https://doi.org/10.1016/j.jeem.2012.02.005
  35. Pereira, M. A., Camanho, A. S., Figueira, J. R., & Marques, R. C. (2021). Incorporating preference information in a range directional composite indicator: The case of Portuguese public hospitals. European Journal of Operational Research. 294(2), 633-650. https://doi.org/10.1016/j.ejor.2021.01.045
  36. Podinovski, V.V., Athanassopoulos, A.D. (1998). Assessing the relative efficiency of decision making units using DEA models with weight restrictions. Journal of the Operational Research Society, 49(5), 500–508. https://doi.org/10.1057/palgrave.jors.2600543
  37. Podinovski, V.V. (2004a). Suitability and redundancy of non-homogeneous weight restrictions for measuring the relative efficiency in DEA. European Journal of Operational Research, 154(2), 380–395.
  38. https://doi.org/10.1016/S0377-2217 (03)00176-0
  39. Podinovski, V. V. (2004b). Production trade-offs and weight restrictions in data envelopment analysis. Journal of the Operational Research Society, 55 (12), 1311–1322. https://doi.org/10.1057/palgrave.jors.2601794
  40. Podinovski, V. V. (2007). Computation of efficient targets in DEA models with production trade-offs and weight restrictions. European Journal of Operational Research, 181(2), 586-591. https://doi.org/10.1016/j.ejor.2006.06.041
  41. Podinovski, V. V., & Bouzdine-Chameeva, T. (2013). Weight restrictions and free production in data envelopment analysis. Operations Research, 61 (2), 426–437.
  42. https://doi.org/10.1287/opre.1120.1122
  43. Podinovski, V.V., Bouzdine-Chameeva, T. (2015). Consistent weight restrictions in data envelopment analysis. European Journal of Operational Research, 244(1), 201–209. https://doi.org/10.1016/j.ejor.2015.01.037
  44. Podinovski, V.V. (2016). Optimal Weights in DEA Models with Weight Restrictions. European Journal of Operational Research. 254(3), 916-924. https://doi.org/10.1016/j.ejor.2016.04.035
  45. Podinovski, V.V. (2016). On single-stage DEA models with weight restrictions. European Journal of Operational Research. 248(3), 1044-1050. https://doi.org/10.1016/j.ejor.2015.07.050
  46. Podinovski, V.V., Wu, J., Argyris, N. (2024). Production trade-offs in models of data envelopment analysis with ratio inputs and outputs: An application to schools in England. European Journal of Operational Research. 313 (1), 359-372.
  47. https://doi.org/10.1016/j.ejor.2015.07.050
  48. Roll, Y., Cook, W.D., Golany, B. (1991). Controlling factor weights in data envelopment analysis. IIE Transactions, 23(1), 2–9. https://doi.org/10.1080/07408179108963835
  49. Ruiz, J. L. (2013). Cross-efficiency evaluation with directional distance functions. European Journal of Operational Research, 228 (1), 181–189. https://doi.org/10.1016/j.ejor.2013.01.030
  50. Sahoo, B. K., Mehdiloozad, M., and Tone, K. (2014). Cost, revenue and profit efficiency measurement in DEA: A directional distance function approach. European Journal of Operational Research, 237 (3), 921–931. https://doi.org/10.1016/j.ejor.2014.02.017
  51. Scheel, H. (2001). Undesirable outputs in efficiency valuations. European Journal of Operational Research, 132, 400–410.
  52. https://doi.org/10.1016/S0377-2217 (00)00160-0
  53. Shen, Z., Balezentis, T., and Streimikis, J. (2022). Capacity utilization and energy-related GHG emission in the European agriculture: A data envelopment analysis approach. Journal of Environmental Management, 318, Article 115517. https://doi.org/10.1016/j.jenvman.2022.115517
  54. Soleimani-damanehab, M., Korhonen, P., and Wallenius, J. (2014). On value efficiency. Optimization, 63 (4), 617–631. https://doi.org/10.1007/978-1-4899-7528-7_10
  55. Song, M., An, Q., Zhang, W., Wang, Z., and Wu, J. (2012). Environmental efficiency evaluation based on data envelopment analysis: A review. Renewable and Sustainable Energy Reviews, 16, 4 465–4 469. https://doi.org/10.1016/j.rser.2012.04.052
  56. Sueyoshi, T., and Goto, M. (2012). DEA radial and non-radial models for unified efficiency under natural and managerial disposability: Theoretical extension by strong complementary slackness conditions. Energy Economics, 34, 700–713. https://doi.org/10.1016/j.eneco.2011.12.013
  57. Thanassoulis, E., Portela, M.C.S., Despi´c, O. (2008). Data envelopment analysis: The mathematical programming approach to efficiency analysis. In H.O. Fried, C.A.K. Lovell, S.S. Schmidt (Eds.), The measurement of productive efficiency and productivity growth (pp. 251–420). New York: Oxford University Press. https://doi.org/10.1093/acprof:oso/9780195183528.003.0003
  58. Wang, K., Xian, Y., Lee, C. Y., Wei, Y. M., and Huang, Z. (2019). On selecting directions for directional distance functions in a non-parametric framework: A review. Annals of Operations Research, 278 (1–2), 43–76. https://doi.org/10.1007/s10479-017-2423-5

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