American Institute of Mathematical Sciences

January  2017, 13(1): 237-249. doi: 10.3934/jimo.2016014

Efficiency measures in fuzzy data envelopment analysis with common weights

 1 Department of International Business, Kao Yuan University, Kaohsiung, 82151, Taiwan 2 Department of Mechanical and Automation Engineering, Ⅰ-Shou University, Kaohsiung, 84001, Taiwan 3 Department of Applied Mathematics, National Chiayi University, Chiayi, 60004, Taiwan

Received  January 2015 Revised  September 2015 Published  March 2016

This work considers providing a common base for measuring the relative efficiency for all the decision-making units (DMUs) with multiple fuzzy inputs and outputs under the fuzzy data envelopment analysis (DEA) framework. It is shown that the fuzzy DEA model with common weights can be reduced into an auxiliary bi-objective fuzzy optimization problem by considering the most and the least favorable conditions simultaneously. An algorithm with the implementation issue for finding the compromise solution of the fuzzy DEA program is developed. A numerical example is included for illustration and comparison purpose. Our results show that the proposed approach is able to provide decision makers the flexibility in measuring the relative efficiency for DMUs with fuzzy inputs and outputs, which not only differentiates efficient units on a common base but also detects some abnormal efficiency scores calculated from other existing methods.

Citation: Cheng-Kai Hu, Fung-Bao Liu, Cheng-Feng Hu. Efficiency measures in fuzzy data envelopment analysis with common weights. Journal of Industrial & Management Optimization, 2017, 13 (1) : 237-249. doi: 10.3934/jimo.2016014
References:
 [1] A. Charnes and W. W. Cooper, The non-Archimedean CCR ratio for efficiency analysis: A rejoinder to Boyd and Fare, European Journal of Operational Research, 15 (1984), 333-334. doi: 10.1016/0377-2217(84)90102-4. Google Scholar [2] W. W. Cooper, K. S. Park and G. Yu, IDEA and AR-IDEA: Models for dealing with imprecise data in DEA, Management Science, 45 (1999), 597-607. doi: 10.1287/mnsc.45.4.597. Google Scholar [3] D. K. Despotis and Y. G. Smirlis, Data envelopment analysis with imprecise data, European Journal of Operational Research, 140 (2002), 24-36. doi: 10.1016/S0377-2217(01)00200-4. Google Scholar [4] R. Green and J. Doyle, Improving discernment in DEA using profiling: A comment, Omega, 24 (1995), 365-366. doi: 10.1016/0305-0483(96)86991-X. Google Scholar [5] A. Hatami-Marbini, A. Emrouznejad and M. Tavana, A taxonomy and review of the fuzzy data envelopment analysis literature: Two decades in the making, European Journal of Operational Research, 214 (2011), 457-472. doi: 10.1016/j.ejor.2011.02.001. Google Scholar [6] C. L. Hwang, Y. J. Lai and T. Y. Liu, A new approach for multiple objective decision making, Computers Ops. Res., 20 (1993), 889-899. doi: 10.1016/0305-0548(93)90109-V. Google Scholar [7] C. L. Hwang and K. Yoon, Multiple Attribute Decision Making: Methods and Applications, Springer, Heidelberg, 1981. Google Scholar [8] C. Kao and H.-T. Hung, Data envelopment analysis with common weights: the compromise solution approach, Journal of the Operational Research Society, 56 (2005), 1196-1203. doi: 10.1057/palgrave.jors.2601924. Google Scholar [9] C. Kao and S.-T. Liu, Data envelopment analysis with missing data: an application to university libraries in Taiwan, Journal of the Operational Research Society, 51 (2000), 897-905. Google Scholar [10] Y. -J. Lai and C. -L. Hwang, Fuzzy Multiple Objective Decision Making: Methods and Applications, Springer, Berlin, 1994. doi: 10.1007/978-3-642-57949-3. Google Scholar [11] D. Jones, M. Tamiz and J. Ries, New Developments in Multiple Objective and Goal Programming, Springer-Verlag, Heidelberg, 2010. doi: 10.1007/978-3-642-10354-4. Google Scholar [12] Y. -J. Lai and C. -L. Hwang, Fuzzy Mathematical Programming: Methods and Applications, Springer-Verlag, New York, 1992. doi: 10.1007/978-3-642-48753-8_3. Google Scholar [13] S. Lertworasirikul, S.-C. Fang, H. L. W. Nuttle and J. A. Joines, Fuzzy BCC model for data envelopment analysis, Fuzzy Optimization and Decision Making, 2 (2003), 337-358. doi: 10.1023/B:FODM.0000003953.39947.b4. Google Scholar [14] S. Lertworasirikul, S.-C. Fang, J. A. Joines and H. L. W. Nuttle, Fuzzy data envelopment analysis (DEA): a possibility approach, Fuzzy Sets and Systems, 139 (2003), 379-394. doi: 10.1016/S0165-0114(02)00484-0. Google Scholar [15] M. Ramezani, M. Bashiri and A. C. Atkinson, A goal programming-TOPSIS approach to multiple response optimization using the concepts of non-dominated solutions and prediction intervals, Expert Systems with Applications, 38 (2011), 9557-9563. doi: 10.1016/j.eswa.2011.01.139. Google Scholar [16] T. Roll, W. D. Cook and B. Golany, Controlling factor weights in data envelopment analysis, IIE Trans, 23 (1991), 2-9. doi: 10.1080/07408179108963835. Google Scholar [17] M. Sakawa, Fuzzy Sets and Interactive Multiobjective Optimizations, Plenum Press, New York, 1993. doi: 10.1007/978-1-4899-1633-4. Google Scholar [18] S. M. Saati, A. Memariani and G. R. Jahanshahloo, Efficiency analysis and ranking of DMUs with fuzzy data, Fuzzy Optimization and Decision Making, 3 (2002), 255-267. Google Scholar [19] J. K. Sengupta, A fuzzy systems approach in data envelopment analysis, Computers and Mathematics with Applications, 24 (1992), 259-266. doi: 10.1016/0898-1221(92)90203-T. Google Scholar [20] H. Späth, Mathematical Algorithmsf or Linear Regression, Academic Press, Boston, 1991. Google Scholar [21] R. E. Steuer, Multiple Criteria Optimization: Theory, Computation, and Application, Wiley, New York, 1986. Google Scholar [22] T. J. Stewart, Data envelopment analysis and multiple criteria decision making: A response, Omega, 22 (1994), 205-206. doi: 10.1016/0305-0483(94)90079-5. Google Scholar [23] K. Yoon, A reconciliation among discrete compromise solutions, J. Opl Res. Sot., 38 (1987), 277-286. Google Scholar [24] P. L. Yu, Multiple-Criteria Decision Making: Concepts, Techniques, and Extensions, Plenum Press, New York, 1985. doi: 10.1007/978-1-4684-8395-6. Google Scholar [25] M. Zeleny, Multiple Criteria Decision Making McGraw-Hill, New York, 1982.Google Scholar [26] H. -J. Zimmermann, Fuzzy Set Theory and Its Applications, 2nd edition, Kluwer Academic, Dordrecht, 1991. Google Scholar

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References:
 [1] A. Charnes and W. W. Cooper, The non-Archimedean CCR ratio for efficiency analysis: A rejoinder to Boyd and Fare, European Journal of Operational Research, 15 (1984), 333-334. doi: 10.1016/0377-2217(84)90102-4. Google Scholar [2] W. W. Cooper, K. S. Park and G. Yu, IDEA and AR-IDEA: Models for dealing with imprecise data in DEA, Management Science, 45 (1999), 597-607. doi: 10.1287/mnsc.45.4.597. Google Scholar [3] D. K. Despotis and Y. G. Smirlis, Data envelopment analysis with imprecise data, European Journal of Operational Research, 140 (2002), 24-36. doi: 10.1016/S0377-2217(01)00200-4. Google Scholar [4] R. Green and J. Doyle, Improving discernment in DEA using profiling: A comment, Omega, 24 (1995), 365-366. doi: 10.1016/0305-0483(96)86991-X. Google Scholar [5] A. Hatami-Marbini, A. Emrouznejad and M. Tavana, A taxonomy and review of the fuzzy data envelopment analysis literature: Two decades in the making, European Journal of Operational Research, 214 (2011), 457-472. doi: 10.1016/j.ejor.2011.02.001. Google Scholar [6] C. L. Hwang, Y. J. Lai and T. Y. Liu, A new approach for multiple objective decision making, Computers Ops. Res., 20 (1993), 889-899. doi: 10.1016/0305-0548(93)90109-V. Google Scholar [7] C. L. Hwang and K. Yoon, Multiple Attribute Decision Making: Methods and Applications, Springer, Heidelberg, 1981. Google Scholar [8] C. Kao and H.-T. Hung, Data envelopment analysis with common weights: the compromise solution approach, Journal of the Operational Research Society, 56 (2005), 1196-1203. doi: 10.1057/palgrave.jors.2601924. Google Scholar [9] C. Kao and S.-T. Liu, Data envelopment analysis with missing data: an application to university libraries in Taiwan, Journal of the Operational Research Society, 51 (2000), 897-905. Google Scholar [10] Y. -J. Lai and C. -L. Hwang, Fuzzy Multiple Objective Decision Making: Methods and Applications, Springer, Berlin, 1994. doi: 10.1007/978-3-642-57949-3. Google Scholar [11] D. Jones, M. Tamiz and J. Ries, New Developments in Multiple Objective and Goal Programming, Springer-Verlag, Heidelberg, 2010. doi: 10.1007/978-3-642-10354-4. Google Scholar [12] Y. -J. Lai and C. -L. Hwang, Fuzzy Mathematical Programming: Methods and Applications, Springer-Verlag, New York, 1992. doi: 10.1007/978-3-642-48753-8_3. Google Scholar [13] S. Lertworasirikul, S.-C. Fang, H. L. W. Nuttle and J. A. Joines, Fuzzy BCC model for data envelopment analysis, Fuzzy Optimization and Decision Making, 2 (2003), 337-358. doi: 10.1023/B:FODM.0000003953.39947.b4. Google Scholar [14] S. Lertworasirikul, S.-C. Fang, J. A. Joines and H. L. W. Nuttle, Fuzzy data envelopment analysis (DEA): a possibility approach, Fuzzy Sets and Systems, 139 (2003), 379-394. doi: 10.1016/S0165-0114(02)00484-0. Google Scholar [15] M. Ramezani, M. Bashiri and A. C. Atkinson, A goal programming-TOPSIS approach to multiple response optimization using the concepts of non-dominated solutions and prediction intervals, Expert Systems with Applications, 38 (2011), 9557-9563. doi: 10.1016/j.eswa.2011.01.139. Google Scholar [16] T. Roll, W. D. Cook and B. Golany, Controlling factor weights in data envelopment analysis, IIE Trans, 23 (1991), 2-9. doi: 10.1080/07408179108963835. Google Scholar [17] M. Sakawa, Fuzzy Sets and Interactive Multiobjective Optimizations, Plenum Press, New York, 1993. doi: 10.1007/978-1-4899-1633-4. Google Scholar [18] S. M. Saati, A. Memariani and G. R. Jahanshahloo, Efficiency analysis and ranking of DMUs with fuzzy data, Fuzzy Optimization and Decision Making, 3 (2002), 255-267. Google Scholar [19] J. K. Sengupta, A fuzzy systems approach in data envelopment analysis, Computers and Mathematics with Applications, 24 (1992), 259-266. doi: 10.1016/0898-1221(92)90203-T. Google Scholar [20] H. Späth, Mathematical Algorithmsf or Linear Regression, Academic Press, Boston, 1991. Google Scholar [21] R. E. Steuer, Multiple Criteria Optimization: Theory, Computation, and Application, Wiley, New York, 1986. Google Scholar [22] T. J. Stewart, Data envelopment analysis and multiple criteria decision making: A response, Omega, 22 (1994), 205-206. doi: 10.1016/0305-0483(94)90079-5. Google Scholar [23] K. Yoon, A reconciliation among discrete compromise solutions, J. Opl Res. Sot., 38 (1987), 277-286. Google Scholar [24] P. L. Yu, Multiple-Criteria Decision Making: Concepts, Techniques, and Extensions, Plenum Press, New York, 1985. doi: 10.1007/978-1-4684-8395-6. Google Scholar [25] M. Zeleny, Multiple Criteria Decision Making McGraw-Hill, New York, 1982.Google Scholar [26] H. -J. Zimmermann, Fuzzy Set Theory and Its Applications, 2nd edition, Kluwer Academic, Dordrecht, 1991. Google Scholar
Input and output data in [2]
 DMU Input Output j Exact Imprecise Exact Ordinal x1j x2jL x2jU y1j y2j 1 100 0.6 0.7 2000 4 2 150 0.8 0.9 1000 2 3 150 1 1 1200 5 4 200 0.7 0.8 900 1 5 200 1 1 600 3
 DMU Input Output j Exact Imprecise Exact Ordinal x1j x2jL x2jU y1j y2j 1 100 0.6 0.7 2000 4 2 150 0.8 0.9 1000 2 3 150 1 1 1200 5 4 200 0.7 0.8 900 1 5 200 1 1 600 3
The positive ideal solution $(E^{\ast}_j)$ and the negative ideal solution $(E^{-}_j)$
 DMUj Ej* Ej− 1 1 2.001 * 10−7 2 0.875 9.01 * 10−8 3 1 1.201 * 10−7 4 1 9.01 * 10−8 5 0.7 6.01 * 10−8
 DMUj Ej* Ej− 1 1 2.001 * 10−7 2 0.875 9.01 * 10−8 3 1 1.201 * 10−7 4 1 9.01 * 10−8 5 0.7 6.01 * 10−8
Efficiency scores and the associated rankings (in parentheses) calculated from different methods
 DMUj Fuzy CCR IDEA approach Model (6) in [3] p = 1 p = 2 p = ∞ 1 1(1) 1(1) 1(1) 1(1) 1(1) 1(1) 2 0.875(4) 0.875(4) 0.875(4) 0.875(4) 0.7825(4) 0.7563(4) 3 1(1) 1(1) 1(1) 1(1) 0.9233(2) 0.9062(2) 4 1(1) 1(1) 1(1) 1(1) 0.9083(3) 0.8943(3) 5 0.7(5) 0.7(5) 0.7(5) 0.7(5) 0.6239(5) 0.6022(5)
 DMUj Fuzy CCR IDEA approach Model (6) in [3] p = 1 p = 2 p = ∞ 1 1(1) 1(1) 1(1) 1(1) 1(1) 1(1) 2 0.875(4) 0.875(4) 0.875(4) 0.875(4) 0.7825(4) 0.7563(4) 3 1(1) 1(1) 1(1) 1(1) 0.9233(2) 0.9062(2) 4 1(1) 1(1) 1(1) 1(1) 0.9083(3) 0.8943(3) 5 0.7(5) 0.7(5) 0.7(5) 0.7(5) 0.6239(5) 0.6022(5)
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