Abstract
Data envelopment analysis (DEA) is a widely used technique for measuring the relative efficiencies of decision-making units (DMUs) with multiple inputs and multiple outputs. The classical DEA models were initially formulated only for desirable inputs and outputs. However, undesirable outputs may be present in the production process which needs to be minimized. In addition, in real-world problems, the observed values of the input and output data are often vague or random. Indeed, decision makers may encounter a hybrid uncertain environment where fuzziness and randomness coexist in a problem. In order to deal with the above problems, this paper proposes fuzzy stochastic DEA model with undesirable outputs. Three fuzzy DEA models with respect to probability–possibility, probability–necessity and probability–credibility constraints are applied. The contribution of this paper is fourfold: (1) the proposed approach considers the impact of undesirable outputs on the performance of DMUs; (2) unlike the existing methods, the proposed solution approach provides efficiency scores in the range of zero and one for all DMUs; (3) the proposed approach analyzes the influence of the presence of both fuzzily imprecision and probabilistic uncertainty in the data over the efficiency results; and (4) a case study in the banking industry is presented to exhibit the efficacy of the procedures and demonstrate the applicability of the proposed model.
Similar content being viewed by others
Notes
This definition is well defined by property monotone of \( {\text{E}}_{k}^{Pos} (\delta ,\gamma ) \) proved in Theorem 3.
References
Bruni, M.E., Conforti, D., Beraldi, P., Tundis, E.: Probabilistically constrained models for efficiency and dominance in DEA. Int. J. Prod. Econ. 117(1), 219–228 (2009)
Charnes, A., Cooper, W.W., Rhodes, E.: Measuring the efficiency of decision making units. Eur. J. Oper. Res. 2(6), 429–444 (1978)
Cooper, W.W., Deng, H., Huang, Z.M., Li, S.X.: Chance constrained programming approaches to technical efficiencies and inefficiencies in stochastic data envelopment analysis. J. Oper. Res. Soc. 53(12), 1347–1356 (2002)
Cooper, W.W., Deng, H., Huang, Z.M., Li, S.X.: Chance constrained programming approaches to congestion in stochastic data envelopment analysis. Eur. J. Oper. Res. 155, 487–501 (2004)
Cooper, W.W., Huang, Z.M., Lelas, V., Li, S.X., Olesen, O.B.: Chance constrained programming formulations for stochastic characterizations of efficiency and dominance in DEA. J. Prod. Anal. 9, 530–579 (1998)
Dubois, D., Prade, H.: Fuzzy Sets and Systems, Theorey and Applications. Academic Press, New York (1980)
Ebrahimnejad, A.: A new link between output-oriented BCC model with fuzzy data in the present of undesirable outputs and MOLP. Fuzzy Inf. Eng. 3, 113–125 (2011)
Ebrahimnejad, A., Tavana, M., Manosourzadeh, S.M.: An interactive MOLP method for solving output-oriented DEA problems with undesirable factors. J. Ind. Manag. Optim. 11(4), 1089–1110 (2015)
Emrouznejad, A., Tavana, M., Hatami-Marbini, A.: The state of the art in fuzzy data envelopment analysis. In: Performance Measurement with Fuzzy Data Envelopment Analysis. published in Studies in Fuzziness and Soft Computing, 309: 1:48, Springer-Verlag (2014)
Fare, R., Grosskopf, S.: Modelling undesirable factors in efficiency evaluation: comment. Eur. J. Oper. Res. 157, 242–245 (2004)
Fare, R., Grosskopf, S., Lovell, C.A.K., Pasurka, C.: Multilateral productivity comparisons when some outputs are undesirable: a nonparametric approach. Rev. Econ. Stat. 71, 90–98 (1989)
Farrokh, M., Azar, A., Jandaghi, G., Ahmadi, A.: A novel robust fuzzy stochastic programming for closed loop supply chain network design under hybrid uncertainty. Fuzzy Sets Syst. (2017). doi:10.1016/j.fss.2017.03.019
Feng, X., Liu, Y.K.: Measurability criteria for fuzzy random vectors. Fuzzy Optim. Decis. Mak. 5, 245–253 (2006)
Guo, P., Tanaka, H., Inuiguchi, M.: Self-organizing fuzzy aggregation models to rank the objects with multiple attributes. IEEE Trans. Syst. Man Cybern. Part A Syst. Hum. 30(5), 573–580 (2000)
Guo, P., Tanaka, H.: Fuzzy DEA: a perceptual evaluation method. Fuzzy Sets Syst. 119(1), 149–160 (2001)
Marbini, Hatami-, Ebrahimnejad, A., Lozano, S.: Fuzzy efficiency measures in data envelopment analysis using lexicographic multi objective approach. Comput. Ind. Eng. 105, 362–376 (2017)
Hatami-Marbini, A., Saati, S., Tavana, M.: An ideal-seeking fuzzy data envelopment analysis framework. Appl. Soft Comput. 10(4), 1062–1070 (2010)
Hatami-Marbini, A., Tavana, M., Ebrahimi, A.: A fully fuzzified data envelopment analysis model. Int. J. Inf. Dec. Sci. 3(3), 252–264 (2011)
Hougaard, J.L.: Fuzzy scores of technical efficiency. Eur. J. Oper. Res. 115(3), 529–541 (1999)
Kao, C., Liu, S.T.: Fuzzy efficiency measures in data envelopment analysis. Fuzzy Sets Syst. 113(3), 427–437 (2000)
Kahraman, C.: Data envelopment analysis using fuzzy concept. In: 28th International symposium on multiple-valued logic, pp. 338–343 (1998)
Tavana, M., Khanjani Shiraz, R., Di Caprio, D.: A chance-constrained portfolio selection model with random-rough variables. Neural Comput. Appl. (2017). doi:10.1007/s00521-017-3014-8
Khanjani, S.R., Tavana, M., Di Caprio, D.: Chance-constrained data envelopment analysis modeling with random-rough data. RAIRO-Op. Res. (2017). doi:10.1051/ro/2016076
Khanjani, S.R., Charles, V., Tavana, M., Di Caprio, D.: A redundancy detection algorithm for fuzzy stochastic multi-objective linear fractional programming problems. Stoch. Anal. Appl. 35(1), 40–62 (2016)
Khanjani Shiraz, R., Tavana, M., Paryab, K.: Fuzzy free disposal hull models under possibility and credibility measures. Int. J. Data Anal. Tech. Strateg. 6(3), 286–306 (2014)
Khanjani Shiraz, R., Charles, V., Jalalzadeh, L.: Fuzzy rough DEA model: a possibility and expected value approaches. Exp. Syst. Appl. 41(2), 434–444 (2014)
Korhonen, P.J., Luptacik, M.: Eco-efficiency analysis of power plants: an extension of data envelopment analysis. Eur. J. Oper. Res. 154, 437–446 (2004)
Kwakernaak, H.: Fuzzy random variables. Part I: definitions and theorems. Inf. Sci. 15(1), 1–29 (1978)
Kwakernaak, H.: Fuzzy random variables. Part II: algorithms and examples for the discrete case. Inf. Sci. 17(3), 253–278 (1979)
Land, K., Lovell, C.A.K., Thore, S.: Chance-constrained data envelopmentanalysis. Manag. Decis. Econ. 14, 541–554 (1994)
Leon, T., Liern, V., Ruiz, J.L., Sirvent, I.: A fuzzy mathematical programming approach to the assessment of efficiency with DEA models. Fuzzy Sets Syst. 139(2), 407–419 (2003)
Lertworasirikul, S., Shu-Cherng, F., Joines, J.A., Nuttle, H.L.W.: Fuzzy data envelopment analysis (DEA): a possibility approach. Fuzzy Sets Syst. 139(2), 379–394 (2003)
Lewis, H.F., Sexton, T.R.: Data envelopment analysis with reverse inputs and outputs. J. Prod. Anal. 21, 113–132 (2004)
Liu, Y., Sumaila, U.R.: Estimating pollution abatement costs of salmon aquaculture: a joint production approach. Land Econ. 86, 569–584 (2010)
Li, S.X.: Stochastic models and variable returns to scales in data envelopment analysis. Eur. J. Oper. Res. 104(3), 532–548 (1998)
Liu, B.: Uncertainty Theory. Springer-Verlag, Berlin (2004)
Liu, B., Liu, Y.K.: Expected value of fuzzy variable and fuzzy expected value models. IEEE Trans. Fuzzy Syst. 10(4), 445–450 (2002)
Liu, Y., Liu, B.: Fuzzy random variable: a scalar expected value operator. Fuzzy Optim. Decis. Mak. 2, 143–160 (2003)
Lu, W.M., Lo, S.F.: A closer look at the economic-environmental disparities for regional development in China. Eur. J. Oper. Res. 183, 882–894 (2007)
Olesen, O.B., Petersen, N.C.: Chance constrained efficiency evaluation. Manage. Sci. 41, 442–457 (1995)
Puri, J., Yadav, S.P.: A fuzzy DEA model with undesirable fuzzy outputs and its application to the banking sector in India. Exp. Syst. Appl. 41(14), 6419–6432 (2014)
Puri, J., Yadav, S.P.: A fully fuzzy DEA approach for cost and revenue efficiency measurements in the presence of undesirable outputs and its application to the banking sector in India. Int. J. Fuzzy Syst. 18(2), 212–226 (2016)
Qin, R., Liu, Y., Liu, Z., Wang, G.: Modeling fuzzy DEA with Type-2 fuzzy variable coefficients, pp. 25–34. Lecture Notes in Computer Science. Springer, Heidelberg (2009)
Qin, R., Liu, Y.K.: A new data envelopment analysis model with fuzzy random inputs and outputs. J. Appl. Math. Comput. 33(1–2), 327–356 (2010)
Qin, R., Liu, Y.K.: Modeling data envelopment analysis by chance method in hybrid uncertain environments. Math. Comput. Simul. 80(5), 922–995 (2010)
Saati, S., Memariani, A., Jahanshahloo, G.R.: Efficiency analysis and ranking of DMUs with fuzzy data. Fuzzy Optim. Decis. Mak. 1, 255–267 (2002)
Sakawa, M.: Fuzzy Sets and Interactive Multiobjective Optimization. Plenum Press, New York (1993)
Sengupta, J.K.: A fuzzy systems approach in data envelopment analysis. Comput. Math Appl. 24(8), 259–266 (1992)
Seiford, M., Zhu, J.: Modeling undesirable factors in efficiency evaluation. Eur. J. Oper. Res. 142, 16–20 (2002)
Sheth, N., Triantis, K.: Measuring and evaluating efficiency and effectiveness using goal programming and data envelopment analysis in a fuzzy environment. Yugoslav J. Oper. Res. 13(1), 35–60 (2003)
Tavana, M., Khanjani Shiraz, R., Hatami-Marbini, A., Agrell, Per J., Paryab, P.: Chance-constrained DEA models with random fuzzy inputs and outputs. Knowl.-Based Syst. 52, 32–52 (2013)
Tavana, M., Khanjani Shiraz, R., Hatami-Marbini, A.: A new chance-constrained DEA model with birandom input and output data. J. Oper. Res. Soc. 65(12), 1824–1839 (2013)
Tavana, M., Khanjani Shiraz, R., Hatami-Marbini, A., Agrell, Per J., Paryab, P.: Fuzzy stochastic data envelopment analysis with application to base realignment and closure (BRAC). Exp. Syst. Appl. 39(15), 12247–12259 (2012)
Tsionas, E.G., Papadakis, E.N.: A bayesian approach to statistical inference in stochastic DEA. Omega 38, 309–314 (2010)
Triantis, K.P., Girod, O.: A mathematical programming approach for measuring technical efficiency in a fuzzy environment. J. Prod. Anal. 10(1), 85–102 (1998)
Triantis, K.: Fuzzy non-radial data envelopment analysis (DEA) measures of technical efficiency in support of an integrated performance measurement system. Int. J. Automot. Technol. Manag. 3(3–4), 328–353 (2003)
Tsolas, J.E., Charles, V.: Incorporating risk into bank efficiency: a satisfying DEA approach to assess the Greek banking crisis. Exp. Syst. Appl. 42(7), 3491–3500 (2015)
Udhayakumar, A., Charles, V., Kumar, M.: Stochastic simulation based genetic algorithm for chance constrained data envelopment analysis problems. Omega 39, 387–397 (2011)
Wang, Y.M., Luo, Y., Liang, L.: Fuzzy data envelopment analysis based upon fuzzy arithmetic with an application to performance assessment of manufacturing enterprises. Exp. Syst. Appl. 36(3), 5205–5211 (2009)
Wu, C., Li, Y., Liu, Q., Wang, K.: A stochastic DEA model considering undesirable outputs with weak disposability. Math. Comput. Model. 58, 980–989 (2012)
Yano, H.: Fuzzy decision making for multi objective stochastic programming problems. Fuzzy Sets Syst. 296, 97–111 (2016)
Zadeh, L.A.: Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst. 1(1), 3–28 (1978)
Zadeh, L.A.: Fuzzy sets. Inf. Control 8(1965), 338–353 (1965)
Zerafat, Angiz L., Emrouznejad, M., Mustafa, A., al-Eraqi, A.S.: Aggregating preference ranking with fuzzy data envelopment analysis. Knowl. Based Syst. 23(6), 512–519 (2010)
Acknowledgements
The authors would like to thank the anonymous reviewers and the editor for their insightful comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Nasseri, S.H., Ebrahimnejad, A. & Gholami, O. Fuzzy Stochastic Data Envelopment Analysis with Undesirable Outputs and its Application to Banking Industry. Int. J. Fuzzy Syst. 20, 534–548 (2018). https://doi.org/10.1007/s40815-017-0367-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40815-017-0367-1