July  2012, 8(3): 623-637. doi: 10.3934/jimo.2012.8.623

A common set of weight approach using an ideal decision making unit in data envelopment analysis

1. 

Department of Mathematics, Tehran-North Branch, Islamic Azad University, P.O. Box 19585-936, Tehran, Iran

2. 

Louvain School of Management, Center of Operations Research and Econometrics (CORE), Université catholique de Louvain, L1.03.01, B-1348 Louvain-la-Neuve, Belgium, Belgium

3. 

Management Information Systems, Lindback Distinguished Chair of Information Systems, La Salle University, Philadelphia, PA19141, United States

Received  September 2011 Revised  January 2012 Published  June 2012

Data envelopment analysis (DEA) is a common non-parametric frontier analysis method. The multiplier framework of DEA allows flexibility in the selection of endogenous input and output weights of decision making units (DMUs) as to cautiously measure their efficiency. The calculation of DEA scores requires the solution of one linear program per DMU and generates an individual set of endogenous weights (multipliers) for each performance dimension. Given the large number of DMUs in real applications, the computational and conceptual complexities are considerable with weights that are potentially zero-valued or incommensurable across units. In this paper, we propose a two-phase algorithm to address these two problems. In the first step, we define an ideal DMU (IDMU) which is a hypothetical DMU consuming the least inputs to secure the most outputs. In the second step, we use the IDMU in a LP model with a small number of constraints to determine a common set of weights (CSW). In the final step of the process, we calculate the efficiency of the DMUs with the obtained CSW. The proposed model is applied to a numerical example and to a case study using panel data from 286 Danish district heating plants to illustrate the applicability of the proposed method.
Citation: Saber Saati, Adel Hatami-Marbini, Per J. Agrell, Madjid Tavana. A common set of weight approach using an ideal decision making unit in data envelopment analysis. Journal of Industrial & Management Optimization, 2012, 8 (3) : 623-637. doi: 10.3934/jimo.2012.8.623
References:
[1]

P. J. Agrell and P. Bogetoft, Economic and environmental efficiency of district heating plants,, Energy Policy, 33 (2005), 1351. doi: 10.1016/j.enpol.2003.12.011. Google Scholar

[2]

P. J. Agrell and P. Bogetoft, Endogenous generalized weights under DEA control,, Working Paper 2010/02, (2010). Google Scholar

[3]

P. Andersen and N. C. Petersen, A procedure for ranking efficient units in data envelopment analysis,, Management Science, 39 (1993), 1261. doi: 10.1287/mnsc.39.10.1261. Google Scholar

[4]

M. Asmild, J. C. Paradi, V. Aggarwal and C. Schaffnit, Combining DEA window analysis with the Malmquist index approach in a study of the Canadian banking industry,, Journal of Productivity Analysis, 21 (2004), 67. doi: 10.1023/B:PROD.0000012453.91326.ec. Google Scholar

[5]

E. Bernroider and V. Stix, A method using weight restrictions in data envelopment analysis for ranking and validity issues in decision making,, Computers and Operations Research, 34 (2007), 2637. doi: 10.1016/j.cor.2005.10.005. Google Scholar

[6]

P. Bogetoft, Incentive efficient production frontiers: An agency perspective on DEA,, Management Science, 40 (1994), 959. doi: 10.1287/mnsc.40.8.959. Google Scholar

[7]

A. Charnes, W. W. Cooper and E. L. Rhodes, Measuring the efficiency of decision making units,, European Journal of Operational Research, 2 (1978), 429. Google Scholar

[8]

C. I. Chiang, M. J. Hwang and Y. H. Liu, Determining a common set of weights in a DEA problem using a separation vector,, Mathematical and Computer Modelling, 54 (2011), 2464. doi: 10.1016/j.mcm.2011.06.002. Google Scholar

[9]

A. Charnes and W. W. Cooper, Programming with linear fractional functions,, Naval Research Logistics Quarterly, 9 (1962), 181. Google Scholar

[10]

A. Charnes, W. W. Cooper, Z. M. Huang and D. B. Sun, Polyhedral cone-ratio DEA models with an illustrative application to large commercial banks,, Journal of Econometrics, 40 (1990), 73. doi: 10.1016/0304-4076(90)90048-X. Google Scholar

[11]

A. Charnes, W. W. Cooper, Q. L. Wei and Z. M. Huang, Cone-ratio data envelopment analysis and multi-objective programming,, International Journal of Systems Sciences, 20 (1989), 1099. doi: 10.1080/00207728908910197. Google Scholar

[12]

W. Cook, M. Kress and L. Seiford, Prioritization models for frontier decision making units in DEA,, European Journal of Operational Research, 59 (1992), 319. doi: 10.1016/0377-2217(92)90148-3. Google Scholar

[13]

D. K. Despotis, Improving the discriminating power of DEA: Focus on globally efficient units,, Journal of the Operational Research Society, 53 (2002), 314. doi: 10.1057/palgrave.jors.2601253. Google Scholar

[14]

L. Friedman and Z. Sinuany-Stern, Scaling units via the canonical correlation analysis in the DEA context,, European Journal of Operational Research, 100 (1997), 629. doi: 10.1016/S0377-2217(97)84108-2. Google Scholar

[15]

F. HosseinzadehLotfi, A. A. Noora, G. R. Jahanshahloo and M. Reshadi, Short communication: One DEA ranking method based on applying aggregate units,, Expert Systems with Applications, 38 (2011), 13468. doi: 10.1016/j.eswa.2011.02.145. Google Scholar

[16]

G. R. Jahanshahloo, F. HosseinzadehLotfi, M. Khanmohammadi, M. Kazemimanesh and V. Rezaie, Ranking of units by positive ideal DMU with common weights,, Expert Systems with Applications, 37 (2010), 7483. doi: 10.1016/j.eswa.2010.04.011. Google Scholar

[17]

G. R. Jahanshahloo, A. Memariani, F. HosseinzadehLotfi and H. Z. Rezai, A note on some of DEA models and finding efficiency and complete ranking using common set of weights,, Applied Mathematics and Computation, 166 (2005), 265. doi: 10.1016/j.amc.2004.04.088. Google Scholar

[18]

G. R. Jahanshahloo, L. Pourkarimi and M. Zarepisheh, Modified MAJ model for ranking decision making units in data envelopment analysis,, Applied Mathematics and Computation, 174 (2006), 1054. doi: 10.1016/j.amc.2005.06.001. Google Scholar

[19]

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. Google Scholar

[20]

M. Khalili, A. S. Camanho, M. C. A. S. Portela and M. R. Alirezaee, The measurement of relative efficiency using data envelopment analysis with assurance regions that link inputs and outputs,, European Journal of Operational Research, 203 (2010), 761. doi: 10.1016/j.ejor.2009.09.002. Google Scholar

[21]

S. Li, G. R. Jahanshahloo and M. Khodabakhshi, A super-efficiency model for ranking efficient units in data envelopment analysis,, Applied Mathematics and Computation, 184 (2007), 638. doi: 10.1016/j.amc.2006.06.063. Google Scholar

[22]

X. B. Li and G. R. Reeves, A multiple criteria approach to data envelopment analysis,, European Journal of Operational Research, 115 (1999), 507. doi: 10.1016/S0377-2217(98)00130-1. Google Scholar

[23]

F. H. F. Liu and H. H. Peng, Ranking of DMUs on the DEA frontier with common weights,, Computers and Operations Research, 35 (2008), 1624. doi: 10.1016/j.cor.2006.09.006. Google Scholar

[24]

F. H. F. Liu and H. H. Peng, A systematic procedure to obtain a preferable and robust ranking of units,, Computers and Operations Research, 36 (2009), 360. Google Scholar

[25]

S. Mehrabian, M. R. Alirezaee and G. R. Jahanshahloo, A complete efficiency ranking of decision making units in DEA,, Computational Optimization and Applications, 14 (1999), 261. doi: 10.1023/A:1008703501682. Google Scholar

[26]

J. C. Paradi, D. N. Reese and D. Rosen, Applications of DEA to measure the efficiency of software production at two large Canadian banks,, Annals of Operations Research, 73 (1997), 91. doi: 10.1023/A:1018953900977. Google Scholar

[27]

N. Ramón, J. L. Ruiz and I. Sirvent, Reducing differences between profiles of weights: A "peer-restricted'' cross-efficiency evaluation,, Omega, 39 (2011), 634. doi: 10.1016/j.omega.2011.01.004. Google Scholar

[28]

Y. Rool, W. D. Cook and B. Golany, Controlling factor weights in data envelopment analysis,, IIE Transactions, 23 (1991), 2. Google Scholar

[29]

S. Saati, Determining a common set of weights in DEA by solving a linear programming, Journal of Industrial Engineering International, 4 (2008), 51. Google Scholar

[30]

S. Saati and A. Memariani, Reducing weight flexibility in fuzzy DEA,, Applied Mathematics and Computation, 161 (2005), 611. doi: 10.1016/j.amc.2003.12.052. Google Scholar

[31]

S. Saati, M. ZarafatAngiz, A. Memariani and G. R. Jahanshahloo, A model for ranking decision making units in data envelopment analysis,, Ricerca Operativa, 31 (2001), 47. Google Scholar

[32]

C. S. Sarrico and R. G. Dyson, Restricting virtual weights in data envelopment analysis,, European Journal of Operational Research, 159 (2004), 17. doi: 10.1016/S0377-2217(03)00402-8. Google Scholar

[33]

T. R. Sexton, R. H. Silkman and A. J. Hogan, Data envelopment analysis: Critique and extensions, in "Measuring Efficiency: An Assessment of Data Envelopment Analysis" (ed. R. H. Silkman),, Jossey-Bass, (1986), 73. Google Scholar

[34]

T. Sueyoshi, DEA nonparametric ranking test and index measurement: Slack-adjusted DEA and an application to Japanese agriculture cooperatives,, Omega, 27 (1999), 315. doi: 10.1016/S0305-0483(98)00057-7. Google Scholar

[35]

R. G. Thompson, P. S. Dharmapala, L. J. Rothenburg and R. M. Thrall, DEA ARs and CRs applied to worldwide major oil companies,, Journal of Productivity Analysis, 5 (1994), 181. Google Scholar

[36]

R. G. Thompson, F. Singleton, R. Thrall and B. Smith, Comparative site evaluations for locating a high-energy physics lab in Texas,, Interfaces, 16 (1986), 35. doi: 10.1287/inte.16.6.35. Google Scholar

[37]

R. M. Thrall, Duality classification and slacks in data envelopment analysis,, Annals of Operation Research, 66 (1996), 109. doi: 10.1007/BF02187297. Google Scholar

[38]

K. Tone, A slacks-based measure of super-efficiency in data envelopment analysis,, European Journal of Operational Research, 143 (2002), 32. doi: 10.1016/S0377-2217(01)00324-1. Google Scholar

[39]

Y.-M. Wang and K.-S. Chin, Discriminating DEA efficient candidates by considering their least relative total scores,, Journal of Computational and Applied Mathematics, 206 (2007), 209. doi: 10.1016/j.cam.2006.06.012. Google Scholar

[40]

Y.-M. Wang and K.-S. Chin, A neutral DEA model for cross-efficiency evaluation and its extension,, Expert Systems with Applications, 37 (2010), 3666. doi: 10.1016/j.eswa.2009.10.024. Google Scholar

[41]

Y.-M. Wang, Y. Luo and Y.-X. Lan, Common weights for fully ranking decision making units by regression analysis,, Expert Systems with Applications, 38 (2011), 9122. doi: 10.1016/j.eswa.2011.01.004. Google Scholar

[42]

Y.-M. Wang, Y. Luo and L. Liang, Ranking decision making units by imposing a minimum weight restriction in the data envelopment analysis,, Journal of Computational and Applied Mathematics, 223 (2009), 469. doi: 10.1016/j.cam.2008.01.022. Google Scholar

show all references

References:
[1]

P. J. Agrell and P. Bogetoft, Economic and environmental efficiency of district heating plants,, Energy Policy, 33 (2005), 1351. doi: 10.1016/j.enpol.2003.12.011. Google Scholar

[2]

P. J. Agrell and P. Bogetoft, Endogenous generalized weights under DEA control,, Working Paper 2010/02, (2010). Google Scholar

[3]

P. Andersen and N. C. Petersen, A procedure for ranking efficient units in data envelopment analysis,, Management Science, 39 (1993), 1261. doi: 10.1287/mnsc.39.10.1261. Google Scholar

[4]

M. Asmild, J. C. Paradi, V. Aggarwal and C. Schaffnit, Combining DEA window analysis with the Malmquist index approach in a study of the Canadian banking industry,, Journal of Productivity Analysis, 21 (2004), 67. doi: 10.1023/B:PROD.0000012453.91326.ec. Google Scholar

[5]

E. Bernroider and V. Stix, A method using weight restrictions in data envelopment analysis for ranking and validity issues in decision making,, Computers and Operations Research, 34 (2007), 2637. doi: 10.1016/j.cor.2005.10.005. Google Scholar

[6]

P. Bogetoft, Incentive efficient production frontiers: An agency perspective on DEA,, Management Science, 40 (1994), 959. doi: 10.1287/mnsc.40.8.959. Google Scholar

[7]

A. Charnes, W. W. Cooper and E. L. Rhodes, Measuring the efficiency of decision making units,, European Journal of Operational Research, 2 (1978), 429. Google Scholar

[8]

C. I. Chiang, M. J. Hwang and Y. H. Liu, Determining a common set of weights in a DEA problem using a separation vector,, Mathematical and Computer Modelling, 54 (2011), 2464. doi: 10.1016/j.mcm.2011.06.002. Google Scholar

[9]

A. Charnes and W. W. Cooper, Programming with linear fractional functions,, Naval Research Logistics Quarterly, 9 (1962), 181. Google Scholar

[10]

A. Charnes, W. W. Cooper, Z. M. Huang and D. B. Sun, Polyhedral cone-ratio DEA models with an illustrative application to large commercial banks,, Journal of Econometrics, 40 (1990), 73. doi: 10.1016/0304-4076(90)90048-X. Google Scholar

[11]

A. Charnes, W. W. Cooper, Q. L. Wei and Z. M. Huang, Cone-ratio data envelopment analysis and multi-objective programming,, International Journal of Systems Sciences, 20 (1989), 1099. doi: 10.1080/00207728908910197. Google Scholar

[12]

W. Cook, M. Kress and L. Seiford, Prioritization models for frontier decision making units in DEA,, European Journal of Operational Research, 59 (1992), 319. doi: 10.1016/0377-2217(92)90148-3. Google Scholar

[13]

D. K. Despotis, Improving the discriminating power of DEA: Focus on globally efficient units,, Journal of the Operational Research Society, 53 (2002), 314. doi: 10.1057/palgrave.jors.2601253. Google Scholar

[14]

L. Friedman and Z. Sinuany-Stern, Scaling units via the canonical correlation analysis in the DEA context,, European Journal of Operational Research, 100 (1997), 629. doi: 10.1016/S0377-2217(97)84108-2. Google Scholar

[15]

F. HosseinzadehLotfi, A. A. Noora, G. R. Jahanshahloo and M. Reshadi, Short communication: One DEA ranking method based on applying aggregate units,, Expert Systems with Applications, 38 (2011), 13468. doi: 10.1016/j.eswa.2011.02.145. Google Scholar

[16]

G. R. Jahanshahloo, F. HosseinzadehLotfi, M. Khanmohammadi, M. Kazemimanesh and V. Rezaie, Ranking of units by positive ideal DMU with common weights,, Expert Systems with Applications, 37 (2010), 7483. doi: 10.1016/j.eswa.2010.04.011. Google Scholar

[17]

G. R. Jahanshahloo, A. Memariani, F. HosseinzadehLotfi and H. Z. Rezai, A note on some of DEA models and finding efficiency and complete ranking using common set of weights,, Applied Mathematics and Computation, 166 (2005), 265. doi: 10.1016/j.amc.2004.04.088. Google Scholar

[18]

G. R. Jahanshahloo, L. Pourkarimi and M. Zarepisheh, Modified MAJ model for ranking decision making units in data envelopment analysis,, Applied Mathematics and Computation, 174 (2006), 1054. doi: 10.1016/j.amc.2005.06.001. Google Scholar

[19]

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. Google Scholar

[20]

M. Khalili, A. S. Camanho, M. C. A. S. Portela and M. R. Alirezaee, The measurement of relative efficiency using data envelopment analysis with assurance regions that link inputs and outputs,, European Journal of Operational Research, 203 (2010), 761. doi: 10.1016/j.ejor.2009.09.002. Google Scholar

[21]

S. Li, G. R. Jahanshahloo and M. Khodabakhshi, A super-efficiency model for ranking efficient units in data envelopment analysis,, Applied Mathematics and Computation, 184 (2007), 638. doi: 10.1016/j.amc.2006.06.063. Google Scholar

[22]

X. B. Li and G. R. Reeves, A multiple criteria approach to data envelopment analysis,, European Journal of Operational Research, 115 (1999), 507. doi: 10.1016/S0377-2217(98)00130-1. Google Scholar

[23]

F. H. F. Liu and H. H. Peng, Ranking of DMUs on the DEA frontier with common weights,, Computers and Operations Research, 35 (2008), 1624. doi: 10.1016/j.cor.2006.09.006. Google Scholar

[24]

F. H. F. Liu and H. H. Peng, A systematic procedure to obtain a preferable and robust ranking of units,, Computers and Operations Research, 36 (2009), 360. Google Scholar

[25]

S. Mehrabian, M. R. Alirezaee and G. R. Jahanshahloo, A complete efficiency ranking of decision making units in DEA,, Computational Optimization and Applications, 14 (1999), 261. doi: 10.1023/A:1008703501682. Google Scholar

[26]

J. C. Paradi, D. N. Reese and D. Rosen, Applications of DEA to measure the efficiency of software production at two large Canadian banks,, Annals of Operations Research, 73 (1997), 91. doi: 10.1023/A:1018953900977. Google Scholar

[27]

N. Ramón, J. L. Ruiz and I. Sirvent, Reducing differences between profiles of weights: A "peer-restricted'' cross-efficiency evaluation,, Omega, 39 (2011), 634. doi: 10.1016/j.omega.2011.01.004. Google Scholar

[28]

Y. Rool, W. D. Cook and B. Golany, Controlling factor weights in data envelopment analysis,, IIE Transactions, 23 (1991), 2. Google Scholar

[29]

S. Saati, Determining a common set of weights in DEA by solving a linear programming, Journal of Industrial Engineering International, 4 (2008), 51. Google Scholar

[30]

S. Saati and A. Memariani, Reducing weight flexibility in fuzzy DEA,, Applied Mathematics and Computation, 161 (2005), 611. doi: 10.1016/j.amc.2003.12.052. Google Scholar

[31]

S. Saati, M. ZarafatAngiz, A. Memariani and G. R. Jahanshahloo, A model for ranking decision making units in data envelopment analysis,, Ricerca Operativa, 31 (2001), 47. Google Scholar

[32]

C. S. Sarrico and R. G. Dyson, Restricting virtual weights in data envelopment analysis,, European Journal of Operational Research, 159 (2004), 17. doi: 10.1016/S0377-2217(03)00402-8. Google Scholar

[33]

T. R. Sexton, R. H. Silkman and A. J. Hogan, Data envelopment analysis: Critique and extensions, in "Measuring Efficiency: An Assessment of Data Envelopment Analysis" (ed. R. H. Silkman),, Jossey-Bass, (1986), 73. Google Scholar

[34]

T. Sueyoshi, DEA nonparametric ranking test and index measurement: Slack-adjusted DEA and an application to Japanese agriculture cooperatives,, Omega, 27 (1999), 315. doi: 10.1016/S0305-0483(98)00057-7. Google Scholar

[35]

R. G. Thompson, P. S. Dharmapala, L. J. Rothenburg and R. M. Thrall, DEA ARs and CRs applied to worldwide major oil companies,, Journal of Productivity Analysis, 5 (1994), 181. Google Scholar

[36]

R. G. Thompson, F. Singleton, R. Thrall and B. Smith, Comparative site evaluations for locating a high-energy physics lab in Texas,, Interfaces, 16 (1986), 35. doi: 10.1287/inte.16.6.35. Google Scholar

[37]

R. M. Thrall, Duality classification and slacks in data envelopment analysis,, Annals of Operation Research, 66 (1996), 109. doi: 10.1007/BF02187297. Google Scholar

[38]

K. Tone, A slacks-based measure of super-efficiency in data envelopment analysis,, European Journal of Operational Research, 143 (2002), 32. doi: 10.1016/S0377-2217(01)00324-1. Google Scholar

[39]

Y.-M. Wang and K.-S. Chin, Discriminating DEA efficient candidates by considering their least relative total scores,, Journal of Computational and Applied Mathematics, 206 (2007), 209. doi: 10.1016/j.cam.2006.06.012. Google Scholar

[40]

Y.-M. Wang and K.-S. Chin, A neutral DEA model for cross-efficiency evaluation and its extension,, Expert Systems with Applications, 37 (2010), 3666. doi: 10.1016/j.eswa.2009.10.024. Google Scholar

[41]

Y.-M. Wang, Y. Luo and Y.-X. Lan, Common weights for fully ranking decision making units by regression analysis,, Expert Systems with Applications, 38 (2011), 9122. doi: 10.1016/j.eswa.2011.01.004. Google Scholar

[42]

Y.-M. Wang, Y. Luo and L. Liang, Ranking decision making units by imposing a minimum weight restriction in the data envelopment analysis,, Journal of Computational and Applied Mathematics, 223 (2009), 469. doi: 10.1016/j.cam.2008.01.022. Google Scholar

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