An uncertain allocation model for data envelopment analysis: A case in the Iranian stock market

Document Type : Article


1 Department of Mathematics, Central Tehran Branch, Islamic Azad University, Tehran, Iran

2 Department of Mathematics, Masjed-Soleiman Branch, Islamic Azad University, Masjed-Soleiman, Iran


Data envelopment analysis (DEA) can be employed for investigating operation and evaluation of units as one of the most important concerns of managers. DEA is a linear programming technique for calculating relative performance of decision-making units (DMUs) with multiple inputs and outputs. Although all inputs and outputs are considered as certain items in these models, there are uncertain items in the real word and existing interference between these two concepts will result in uncertain models.
Allocation models were studied in an uncertain environment with belief degree based uncertain input costs and output prices. Belief degree based uncertainty is useful for cases where there is no historical information on an uncertain event. Utilizing the uncertain entropy model as a second objective function, the cost and revenue models showed an optimal performance with a maximum dispersion rate in their constituent components. As a solution methodology, the uncertain allocation models were separately converted into crisp models by expect value (EV) and expected value and chance-constrained (EVCC) methods. A practical example from the Iranian Stock Market was also presented to evaluate the performance of the new model.


1. Charnes, A., Cooper, W.W., and Rhodes, E. "Measuring the efficiency of decision making units", European Journal of Operational Research, 2, pp. 429-444 (1978).
2. Farrell, M.J. "The measurement of productive efficiency", Journal of the Royal Statistical Society A, 120, pp. 253-281 (1957).
3. Banker, R.D., Charnes, A., and Cooper, W.W. "Some models for estimating technical and scale inefficiencies in data envelopment analysis", Management Science, 30(9), pp. 1031-1142 (1984).
4. Cooper, W.W., Park, K.S., and Pastor, J.T. "RAM: a range adjusted measure of on efficiency for use with additive models, and relations to other models and measures in DEA", Journal of Productivity Analysis, 11, pp. 5-24 (1999).
5. Sueyoshi, T. "DEA non-parametric ranking test and index measurement: Slack-adjust DEA and an application to Japanese agriculture cooperative", Omega, 27, pp. 315-326 (1999).
6. Seiford, L.M. and Thrall, R.M. "Recent developments in DEA, the mathematical programming approach to frontier analysis", Journal of Econometrics, 46, pp. 7-38 (1990).
7. Tone, K. "A slacks-based measure of efficiency in data envelopment analysis", European Journal of Operational Research, 130(3), pp. 498-509 (2001).
8. Deprins, D., Simar, L., and Tulkens, H. "Measuring labor efficiency in post offices", In The Performance of Public Enterprises, M. Marchand, P. Pestieau, Tulkens, H., (Eds), Elsevier Science Publishers, Amesterdam, pp. 243-267 (1984).
9. Sahoo, B.K. and Tone, K. "Non-parametric measurement of economies of scale and scope in noncompetitive environment with price uncertainty", Omega, 41, pp. 97-111 (2013).
10. Fare, R., Grosskopf, S., and Lovell, C.A.K., Measurement of Efficiency of Production, Boston: Kluwer- Nijhoff Publishing Co., Inc. (1985).
11. Jamshidi, M., Sanei, M., Mahmoodirad, A., et al. "Uncertain RUSSEL data envelopment analysis model: A case study in Iranian banks", Journal of Intelligent & Fuzzy Systems, 37(2), pp. 2937-2951 (2019).
12. Sengupta, J.K. "Efficiency measurement in stochastic input-output system", International Journal of Systems Science, 13, pp. 273-287 (1982).
13. Banker, R.D. "Maximum likelihood, consistency and data envelopment analysis. A statistical foundation", Management Science, 39, pp. 1265-1273 (1993).
14. Cooper, W.W., Huang, Z., Lelas, V., et al. "Chanceconstrained programming formulations for stochastic characterizations of efficiency and dominance in DEA", Journal of Productivity Analysis, 9, pp. 53-79 (1998).
15. Grosskopf, S. "Statistical inference and nonparametric efficiency: a selective survey", Journal of Productivity Analysis, 7, pp. 161-176 (1996).
16. Olsen, O. and Petersen, N.C. "Chance constrained efficiency evaluation", Management Science, 131, pp. 442-457 (1995).
17. Cooper, W.W., Park, K.S., and Yu, G. "Idea and AR-IDEA: models for dealing with imprecise data in DEA", Management Science, 45, pp. 597-607 (1999).
18. Cooper, W.W., Park, K.S., and Yu, G. "An illustrative application of idea (imprecise data envelopment analysis) to Korean mobile telecommunication company", Operations Research, 49, pp. 807-820 (2001).
19. Cooper, W.W., Park, K.S., and Yu, G. "IDEA (imprecise data envelopment analysis) with CMDs (column maximum decision making units)", The Journal of the Operational Research Society, 52, pp. 176-181 (2001).
20. Kao, C. and Liu, S. "Fuzzy efficiency measures in data envelopment analysis", Fuzzy Sets and Systems, 113, pp. 427-437 (2000).
21. Entani, T., Maeda, Y., and Tanaka, H. "Dual models of interval DEA and its extension to interval data", European Journal of Operational Research, 136, pp. 32-45 (2002).
22. Zadeh, L.A. "Fuzzy sets as a basis for a theory of possibility", Fuzzy Sets and Systems, 1, pp. 3-28 (1978).
23. Guo, P. and Tanaka, H. "Fuzzy DEA: a perceptual evaluation method", Fuzzy Sets and System, 119, pp. 149-160 (2001).
24. Lertworasirikul, S., Fang, S., Joines, J.A., et al. "Fuzzy data envelopment analysis (DEA): A possibility approach", Fuzzy Sets and Systems, 139, pp. 379-394 (2003).
25. Liu, B., Uncertain Theory, Springer, Berlin, Germany, 2nd Edn. (2007).
26. Jamshidi, M., Sanei, M., Mahmoodirad, A., et al. "Uncertain SBM data envelopment analysis model: A case study in Iranian banks", International Journal of Finance and Economics, 26(2), pp. 2674-2689 (2021).
27. Mahmoodirad, A., Dehghan, R., and Niroomand, S. "Modelling linear fractional transportation problem in belief degree-based uncertain environment", Journal of Experimental & Theoretical Intelligence, 31(3), pp. 393-408 (2018).
28. Liu, B., Uncertain Theory, Springer, Berlin, Germany, 4th Edn. (2015).
29. Wen, M., Guo, L., Kang, R., et al. "Data envelopment analysis with uncertain inputs and outputs", Journal of Applied Mathematics, 2014, pp. 1-7, Article ID 307108 (2014). 307108.
30. Wen, M., Qin, Z., Kang, R., et al., Sensitivity and Stability Analysis of the Additive Model in Uncertain Data Envelopment Analysis, Springer-Verlag Berlin Heidelberg (2014).
31. Wen, M., Zhang, Q., Kang, R., et al. "Some new ranking criteria in data envelopment analysis under uncertain environment", Computer & Industrial Engineering, 110, pp. 498-504 (2017).
32. Lio, W. and Liu, B. "Uncertain data envelopment analysis with imprecisely observed inputs and outputs", Fuzzy Optimization and Decision Making, 17(3), pp. 357-373 (2017).
33. Mahmoodirad, A., Niroomand, S., and Hosseinzadeh Lotfi, F. "An effective solution approach for multiobjective fractional fixed charge problem with fuzzy parameters", Scientia Iranica, 27(4), pp. 2057-2068 (2020).
34. Majumder, S., Kundu, P., Kar, S., et al. "Uncertain multi-objective multi-item fixed charge solid transportation problem with budget constraint", Soft Computing, 23, pp. 3279-3301 (2020).
35. Lio, W. and Liu, B. "Residual and confidence interval for uncertain regression model with imprecise observations", Journal of Intelligent & Fuzzy Systems, 35(2), pp. 2573-2583 (2018).
36. Chen, W., Wang, Y., Gupta, P., et al. "A novel hybrid heuristic algorithm for a new uncertain meanvariance- skewness portfolio selection model with real constraints", Applied Intelligence, 48, pp. 2996-3018 (2018).
37. Zhai, J. and Bai, M. "Mean-variance model for portfolio optimization with background risk based on uncertainty theory", International Journal of General Systems, 47(3), pp. 294-312 (2018).
38. Dalman, H. "Entropy-based multi-item solid transportation problems with uncertain variables", Soft Computing, 23, pp. 5931-5943 (2018).
39. Zhang, B., Peng, J., Li, S., et al. "Fixed charge solid transportation problem in uncertain environment and its algorithm", Computer & Industrial Engineering, 102, pp. 186-197 (2016).
40. Liu, B. "Some research problems in uncertainty theory",Journal of Uncertain System, 3(1), pp. 3-10 (2009).
41. Chen, X. and Dai, W. "Maximum entropy principle for uncertain variables", International Journal of Fuzzy Systems, 13(3), pp. 232-236 (2011).
42. Dai, W. and Chen, X. "Entropy of function of uncertain variables", Mathematical and Computer Modelling, 55(3), pp. 754-760 (2012).
43. Liu, B. "Why is there a need for uncertainty theory", Journal of Uncertain Systems, 6, pp. 3-10 (2012).
44. Peng, Z.X. and Iwamura, K. "Some properties of product uncertain measure", Journal of Uncertain Systems, 6(4), pp. 263-269 (2012).
45. Kolmogorov, A.N., Grundbegrie der Wahrscheinlichkeitsrechnung, Julius Springer, Berlin (1933).
46. Shannon, C.E. and Weaver, W., The Mathematical Theory of Communication, University of Illinois Press, Urbana (1949).
47. Liu, B., Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer Verlag, Berlin (2010).
48. Fubini, G. "Sugli integral multipli", Rom. Acc. L. Rend., 16(1), pp. 608-614 (1907).