Document Type : Article

**Authors**

^{1}
Department of Mathematics, Ardabil Branch, Islamic Azad University, Ardabil, Iran

^{2}
Department of Business Systems and Analytics, Lindback Distinguished Chair of Information Systems and Decision Sciences, La Salle University, Philadelphia, PA 19141, USA.; Department of Business Information Systems, Faculty of Business Administration and Economics, University of Paderborn, D-33098 Paderborn, Germany.

**Abstract**

We present an integrated data envelopment analysis (DEA) and Malmquist productivity index (MPI) to evaluate the performance of decision making units (DMUs) by using a directional distance function with undesirable interval outputs. The MPI calculation is performed to compare the efficiency of the DMUs in distinct time periods. The uncertainty inherent in real-world problems is considered by using the best and worst-case scenarios, defining an interval for the MPI and reflecting the DMUs’ advancement or regress. The optimal solution of the robust model lies in the efficiency interval, i.e., it is always equal to or less than the optimal solution in the optimistic case and equal to or greater than the optimal solution in the pessimistic case. We also present a case study in the banking industry to demonstrate applicability and efficacy of the proposed integrated approach.

**Keywords**

- Data envelopment analysis
- Malmquist productivity index
- Interval approach
- directional distance function
- undesirable outputs

**Main Subjects**

References:

1. Farrell, M.J. "The measurement of productive efficiency", Journal of the Royal Statistical Society, 120(3), pp. 253-281 (1957).

2. Charnes, A., Cooper, W.W., and Rhodes, E. "Measuring the efficiency of decision making units", European Journal of Operational Research, 2(6), pp. 429-444 (1978).

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. 1078-1092 (1984).

4. Pittman, R.W. "Multilateral productivity comparisons with undesirable outputs", Economic Journal, 93(372), pp. 883-891 (1983).

5. Caves, D.W., Christensen, L.R., and Diewert, E. "Multilateral comparisons of output, input and productivity using superlative index numbers", The Economic Journal, 92(365), pp. 73-86 (1982).

6. Ardabili, J.S., Aghayi, N., and Monzali, A.L. "New efficiency using undesirable factors of data envelopment analysis", Modeling & Optimization, 9(2), pp. 249-255 (2007).

7. Malmquist, S. "Index numbers and indifference surfaces", Trabajos de Estatistica, 4(2), pp. 209- 242 (1953).

8. Fare, R., Grosskopf, S., and Logan, J. "The relative efficiency of Illinois electric utilities", Resources and Energy, 5, pp. 349-367 (1983).

9. Soyster, A.L. "Convex programming with set- inclusive constraints and applications to inexact linear programming", Operational Research, 21, pp. 1154-1157 (1972).

10. Seiford, L.M. and Zhu, J. "Modeling undesirable factors inefficiency valuation", European Journal of Operational Research, 142(1), pp. 16-20 (2002).

11. Chambers, R.G., Chung, Y., and Fare, R. "Benefit and distance function", Journal of Economic Theory, 70(2), pp. 407-419 (1996).

12. Chung, Y.H., Fare, R., and Grosskopf, S. "Productivity and undesirable outputs a directional distance function approach", Journal of Environmental Management, 51(3), pp. 229-240 (1997).

13. Shepherd, R.W., Theory of Cost and Production Functions, Princeton, NJ, USA: Princeton University press (1970).

14. Zanella, A., Camanho, A., and Dias, T. "Undesirable outputs and weighting schemes in composite indicators based on data envelopment analysis", European Journal of Operational Research, 245, pp. 517-530 (2015).

15. Iftikhar, Y., Wang, Z., Zhang, B., and Wang, B. "Energy and CO2 emissions efficiency of major economies: A network DEA approach", Energy, 147, pp. 197-207 (2018).

16. Khoshandam, L., Kazemi, R., and Amirteimoori, A. "Marginal rate of substitution in data envelopment analysis with undesirable outputs: A directional approach", Measurement, 68, pp. 49-57 (2015).

17. Barnabe, W. "Disaggregation of the cost Malmquist productivity index with joint and output-specific inputs", Omega, 75, pp. 1-12 (2018).

18. Sueyoshi, T., Goto, M., and Wang, D. "Malmquist index measurement for sustainability enhancement in Chinese municipalities and provinces", Energy Economics, 67, pp. 554-571 (2017).

19. Sueyoshi, T. and Goto, M. "DEA environmental assessment in time horizon: Radial approach Malmquist index measurement on petroleum companies", Energy Economics, 51, pp. 329-345 (2015).

20. Fuentes, R. and Lillo-Banuls, A. "Smoothed bootstrap Malmquist index based on DEA model to compute productivity of tax offices", Expert Systems with Applications, 42, pp. 2442-2450 (2015).

21. Yu, C., Shi, L., Wang, Y., Chang, Y., and Cheng, B. "The eco-efficiency of pulp and paper industry in China: an assessment based on slacks-based measure and Malquist-Luenberger index", Journal of Cleaner Production, 127, pp. 511-521 (2016).

22. Kao, C. "Measurement and decomposition of the Malmquist productivity index for parallel production systems", Omega, 67, pp. 54-59 (2016).

23. Maroto, A. and Zofio, J. "Accessibility gains and road transport infrastructure in Spain: A productivity approach based on the Malmquist index", Journal of Transport Geography, 52, pp. 143-152 (2016).

24. Emrouznejad, A., Rostamy-Malkhalifeh, M., Hatami-Marbini, A., Tavana, M., and Aghayi, N. "An overall profit Malmquist productivity index with fuzzy and interval data", Mathematical and Computer Modelling, 54(11-12), pp. 2827-2838 (2011).

25. Wanke, P., Barros, C.P., and Emrouznejad, A. "Assessing productive efficiency of banks using integrated Fuzzy-DEA and bootstrapping a case of Mozambican banks", European Journal of Operational Research, 249(1), pp. 378-389 (2016).

26. Mashayekhi, Z. and Omrani, H. "An integrated multiobjective Markowitz-DEA cross efficiency model with fuzzy returns for portfolio selection problem", Operation Research, 38, pp. 1-9 (2016).

27. Aghayi, N. "Cost efficiency measurement with fuzzy data in DEA", Journal of Intelligent and Fuzzy Systems, 32, pp. 409-420 (2017).

28. Toloo, M., Aghayi, N., and Rostamy-Malkhalifeh, M. "Measuring overall profit efficiency with interval data", Applied Mathematics and Computation, 201(1-2), pp. 640-649 (2008).

29. Hatami-Marbini, A., Emrouznejad, A., and Agrell,P. "Interval data without sign restrictions in DEA", Applied Mathematical Modelling, 38(7-8), pp. 2028-2036 (2014).

30. Salehpour, S. and Aghayi, N. "The most revenue efficiency with price uncertainty", International Journal of Data Envelopment Analysis, 3, pp. 575-592 (2015).

31. Kouvelis, P. and Yu, G., Robust Discrete Optimization and Its Applications, Kluwer Academic publishers Norwell, MA (1997).

32. Ben-Tall, A. and Nemirovski, A. "Robust convex optimization", Mathematical Operation Research, 23, pp. 769-805 (1998).

33. El-Ghaoui, L. and Lebret, H. "Robust solutions to least-squares problems to uncertain data matrices", Sima Journal on Matrix Analysis and Applications, 18, pp. 1035-1064 (1997).

34. Bertsimas, D. and Sim, M. "The price of the robustness", Operation Research, 52, pp. 35-53 (2004).

35. Zahedi-Seresht, M., Jahanshahloo, G.R., and Jablonsky, J. "A robust data envelopment analysis model with different scenarios", Applied Mathematical Modelling, 52, pp. 306-319 (2017).

36. Yousefi, S., Soltani, R., Saen, R.F., and Pishvaee, M.S. "A robust fuzzy possibilistic programming for a new network GP-DEA model to evaluate sustainable supply chains", Journal of Cleaner Production, 166, pp. 537-549 (2017).

37. Chung-Cheng, L. "Robust data envelopment analyses approaches for evaluating algorithmic performance", Computers and Industrial Engineering, 81, pp. 78-89 (2015).

38. Mardani, M. and Salarpour, M. "Measuring technical efficiency of potato production in Iran using robust data envelopment analysis", Information Processing in Agriculture, 2(1), pp. 6-14 (2015).

39. Aghayi, N., Tavana, M., and Raayatpanah, M.A. "Robust efficiency measurement with common set of weights under varying degrees of conservatism and data uncertainty", European Journal of Industrial Engineering, 10(30), pp. 385-405 (2016).

40. Aghayi, N. and Maleki, B. "Efficiency measurement of DMUs with undesirable outputs under uncertainty based on the directional distance function: Application on Bank Industry", Energy, 112, pp. 376-387 (2016).

41. Ray, C. and Desli, E. "Productivity growth, technical progress, and efficiency change in industrialized countries: comment", The American Economic Review, 87, pp. 1033-1039 (1997).

2. Charnes, A., Cooper, W.W., and Rhodes, E. "Measuring the efficiency of decision making units", European Journal of Operational Research, 2(6), pp. 429-444 (1978).

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. 1078-1092 (1984).

4. Pittman, R.W. "Multilateral productivity comparisons with undesirable outputs", Economic Journal, 93(372), pp. 883-891 (1983).

5. Caves, D.W., Christensen, L.R., and Diewert, E. "Multilateral comparisons of output, input and productivity using superlative index numbers", The Economic Journal, 92(365), pp. 73-86 (1982).

6. Ardabili, J.S., Aghayi, N., and Monzali, A.L. "New efficiency using undesirable factors of data envelopment analysis", Modeling & Optimization, 9(2), pp. 249-255 (2007).

7. Malmquist, S. "Index numbers and indifference surfaces", Trabajos de Estatistica, 4(2), pp. 209- 242 (1953).

8. Fare, R., Grosskopf, S., and Logan, J. "The relative efficiency of Illinois electric utilities", Resources and Energy, 5, pp. 349-367 (1983).

9. Soyster, A.L. "Convex programming with set- inclusive constraints and applications to inexact linear programming", Operational Research, 21, pp. 1154-1157 (1972).

10. Seiford, L.M. and Zhu, J. "Modeling undesirable factors inefficiency valuation", European Journal of Operational Research, 142(1), pp. 16-20 (2002).

11. Chambers, R.G., Chung, Y., and Fare, R. "Benefit and distance function", Journal of Economic Theory, 70(2), pp. 407-419 (1996).

12. Chung, Y.H., Fare, R., and Grosskopf, S. "Productivity and undesirable outputs a directional distance function approach", Journal of Environmental Management, 51(3), pp. 229-240 (1997).

13. Shepherd, R.W., Theory of Cost and Production Functions, Princeton, NJ, USA: Princeton University press (1970).

14. Zanella, A., Camanho, A., and Dias, T. "Undesirable outputs and weighting schemes in composite indicators based on data envelopment analysis", European Journal of Operational Research, 245, pp. 517-530 (2015).

15. Iftikhar, Y., Wang, Z., Zhang, B., and Wang, B. "Energy and CO2 emissions efficiency of major economies: A network DEA approach", Energy, 147, pp. 197-207 (2018).

16. Khoshandam, L., Kazemi, R., and Amirteimoori, A. "Marginal rate of substitution in data envelopment analysis with undesirable outputs: A directional approach", Measurement, 68, pp. 49-57 (2015).

17. Barnabe, W. "Disaggregation of the cost Malmquist productivity index with joint and output-specific inputs", Omega, 75, pp. 1-12 (2018).

18. Sueyoshi, T., Goto, M., and Wang, D. "Malmquist index measurement for sustainability enhancement in Chinese municipalities and provinces", Energy Economics, 67, pp. 554-571 (2017).

19. Sueyoshi, T. and Goto, M. "DEA environmental assessment in time horizon: Radial approach Malmquist index measurement on petroleum companies", Energy Economics, 51, pp. 329-345 (2015).

20. Fuentes, R. and Lillo-Banuls, A. "Smoothed bootstrap Malmquist index based on DEA model to compute productivity of tax offices", Expert Systems with Applications, 42, pp. 2442-2450 (2015).

21. Yu, C., Shi, L., Wang, Y., Chang, Y., and Cheng, B. "The eco-efficiency of pulp and paper industry in China: an assessment based on slacks-based measure and Malquist-Luenberger index", Journal of Cleaner Production, 127, pp. 511-521 (2016).

22. Kao, C. "Measurement and decomposition of the Malmquist productivity index for parallel production systems", Omega, 67, pp. 54-59 (2016).

23. Maroto, A. and Zofio, J. "Accessibility gains and road transport infrastructure in Spain: A productivity approach based on the Malmquist index", Journal of Transport Geography, 52, pp. 143-152 (2016).

24. Emrouznejad, A., Rostamy-Malkhalifeh, M., Hatami-Marbini, A., Tavana, M., and Aghayi, N. "An overall profit Malmquist productivity index with fuzzy and interval data", Mathematical and Computer Modelling, 54(11-12), pp. 2827-2838 (2011).

25. Wanke, P., Barros, C.P., and Emrouznejad, A. "Assessing productive efficiency of banks using integrated Fuzzy-DEA and bootstrapping a case of Mozambican banks", European Journal of Operational Research, 249(1), pp. 378-389 (2016).

26. Mashayekhi, Z. and Omrani, H. "An integrated multiobjective Markowitz-DEA cross efficiency model with fuzzy returns for portfolio selection problem", Operation Research, 38, pp. 1-9 (2016).

27. Aghayi, N. "Cost efficiency measurement with fuzzy data in DEA", Journal of Intelligent and Fuzzy Systems, 32, pp. 409-420 (2017).

28. Toloo, M., Aghayi, N., and Rostamy-Malkhalifeh, M. "Measuring overall profit efficiency with interval data", Applied Mathematics and Computation, 201(1-2), pp. 640-649 (2008).

29. Hatami-Marbini, A., Emrouznejad, A., and Agrell,P. "Interval data without sign restrictions in DEA", Applied Mathematical Modelling, 38(7-8), pp. 2028-2036 (2014).

30. Salehpour, S. and Aghayi, N. "The most revenue efficiency with price uncertainty", International Journal of Data Envelopment Analysis, 3, pp. 575-592 (2015).

31. Kouvelis, P. and Yu, G., Robust Discrete Optimization and Its Applications, Kluwer Academic publishers Norwell, MA (1997).

32. Ben-Tall, A. and Nemirovski, A. "Robust convex optimization", Mathematical Operation Research, 23, pp. 769-805 (1998).

33. El-Ghaoui, L. and Lebret, H. "Robust solutions to least-squares problems to uncertain data matrices", Sima Journal on Matrix Analysis and Applications, 18, pp. 1035-1064 (1997).

34. Bertsimas, D. and Sim, M. "The price of the robustness", Operation Research, 52, pp. 35-53 (2004).

35. Zahedi-Seresht, M., Jahanshahloo, G.R., and Jablonsky, J. "A robust data envelopment analysis model with different scenarios", Applied Mathematical Modelling, 52, pp. 306-319 (2017).

36. Yousefi, S., Soltani, R., Saen, R.F., and Pishvaee, M.S. "A robust fuzzy possibilistic programming for a new network GP-DEA model to evaluate sustainable supply chains", Journal of Cleaner Production, 166, pp. 537-549 (2017).

37. Chung-Cheng, L. "Robust data envelopment analyses approaches for evaluating algorithmic performance", Computers and Industrial Engineering, 81, pp. 78-89 (2015).

38. Mardani, M. and Salarpour, M. "Measuring technical efficiency of potato production in Iran using robust data envelopment analysis", Information Processing in Agriculture, 2(1), pp. 6-14 (2015).

39. Aghayi, N., Tavana, M., and Raayatpanah, M.A. "Robust efficiency measurement with common set of weights under varying degrees of conservatism and data uncertainty", European Journal of Industrial Engineering, 10(30), pp. 385-405 (2016).

40. Aghayi, N. and Maleki, B. "Efficiency measurement of DMUs with undesirable outputs under uncertainty based on the directional distance function: Application on Bank Industry", Energy, 112, pp. 376-387 (2016).

41. Ray, C. and Desli, E. "Productivity growth, technical progress, and efficiency change in industrialized countries: comment", The American Economic Review, 87, pp. 1033-1039 (1997).

Transactions on Industrial Engineering (E)

November and December 2019Pages 3819-3834