American Institute of Mathematical Sciences

Advanced
March  2020, 16(2): 633-651. doi: 10.3934/jimo.2018171

A new method for ranking decision making units using common set of weights: A developed criterion

Gholam Hassan Shirdel , and  Somayeh Ramezani-Tarkhorani

Department of Mathematics, Faculty of Sciences, University of Qom, Alghadir Bld., Postal code:3716146611, Qom, Iran

* Corresponding author: Gholam Hassan Shirdel

Received  December 2016 Revised  June 2017 Published  October 2018

In this paper we have developed a new model by altering Liu and Peng's approach [20] toward ranking method using CSW. In fact, we have adopted a new criterion which is stronger in terms of maximizing efficiencies. After showing advantages of our model theoretically and illustrating it geometrically, two examples demonstrated how the proposed method is practically more capable.

Keywords: Data envelopment analysis, ranking, common set of weights, linear programming, efficiency.
Mathematics Subject Classification: Primary: 90C05, 90C29; Secondary: 90B50.
Citation: Gholam Hassan Shirdel, Somayeh Ramezani-Tarkhorani. A new method for ranking decision making units using common set of weights: A developed criterion. Journal of Industrial & Management Optimization, 2020, 16 (2) : 633-651. doi: 10.3934/jimo.2018171
References:
[1]

N. Adler, L. Friedman and Z. Sinuany-Stern, Review of ranking methods in the data envelopment analysis context, Eur J Oper Res, 140 (2002), 249-265.  doi: 10.1016/S0377-2217(02)00068-1.  Google Scholar

[2]

N. Aghayi, M. Tavana and M. A. Raayatpanah, Robust efficiency measurement with common set of weights under varying degrees of conservatism and data uncertainty, Eur J. Ind Eng, 10 (2016), 385-405.  doi: 10.1504/EJIE.2016.076386.  Google Scholar

[3]

A. Aldamak and S. Zolfaghari, Review of efficiency ranking methods in data envelopment analysis, Meas, 106 (2017), 161-172.  doi: 10.1016/j.measurement.2017.04.028.  Google Scholar

[4]

R. D. Banker, A. Charnes and W. W. Cooper, Some models for estimating technical and scale inefficiencies in data envelopment analysis, Manage Sci, 30 (1984), 1031-1142.  doi: 10.1287/mnsc.30.9.1078.  Google Scholar

[5]

A. Charnes, W. W. Cooper and E. Rhodes, Measuring the efficiency of DMUs, Eur J Oper Res, 2 (1978), 429-444.  doi: 10.1016/0377-2217(78)90138-8.  Google Scholar

[6]

C. I. Chiang, M. J. Hwang and Y. H. Liu, Determining a common set of weights in a DEA problem using a separation vector, Math and Comput Model, 54 (2002), 2464-2470.  doi: 10.1016/j.mcm.2011.06.002.  Google Scholar

[7]

C. I. Chiang and G. H. Tzeng, A new efficiency measure for DEA: Efficiency achievement measure established on fuzzy multiple objectives programming, J Manage, 17 (2000), 369-388.   Google Scholar

[8]

W. D. Cook, Y. Roll and A. Kazakov, A DEA model for measuring the relative efficiency of highway maintenance patrols, Inform Syst Oper Res, 28 (1990), 113-124.   Google Scholar

[9]

D. K. Despotis, Improving the discriminating power of DEA: Focus on globally efficient units, J Oper Res Soc, 53 (2002), 314-323.  doi: 10.1057/palgrave.jors.2601253.  Google Scholar

[10]

J. A. Ganley and S. A. Cubbin, Public Sector Efficiency Measurement: Applications of Data Envelopment Analysis, North-Holland, Amsterdam, 1992. Google Scholar

[11]

A. Hatami Marbini, M. Tavana, P. J. Agrell, F. Hosseinzadeh Lotfi and Z. Ghelej Beigi, A common-weights DEA model for centralized resource reduction and target setting, Comput Ind Eng, 79 (2015), 195-203.   Google Scholar

[12]

F. Hosseinzadeh Lotfi, M. Rostamy-Malkhlifeh, G. R. Jahanshahloo, Z. Moghaddas, M. Khodabakhshi and M. Vaez-Ghasemi, A review of ranking models in data envelopment analysis, J Appl Math., 2013 (2013), Article ID: 492421, 20 pages. Google Scholar

[13]

C. K. Hu, F. B. Liu and C. F. Hu, Efficiency measures in fuzzy data envelopment analysis with common weights, J Ind Manage Opt, 13 (2017), 237-249.  doi: 10.3934/jimo.2016014.  Google Scholar

[14]

C. L. Hwang and A. S. M. Masud, Multiple Objective Decision Making: Methods and Applications, Springer, Berlin, 1979. Google Scholar

[15]

G. R. Jahanshahloo, F. Hosseinzadeh Lotfi, M. Khanmohammadi, M. Kazemimanesh and V. Rezaie, Ranking of units by positive ideal DMU with common weights, Expert Syst Appl, 37 (2010), 7483-7488.  doi: 10.1016/j.eswa.2010.04.011.  Google Scholar

[16]

C. Kao and H. Hung, Data envelopment analysis with common weights: The comprise solution approach, J Oper Res Soc, 56 (2005), 1196-1203.  doi: 10.1057/palgrave.jors.2601924.  Google Scholar

[17]

M. N. Kritikos, A full ranking methodology in data envelopment analysis based on a set of dummy decision making units, Expert Systems with Applications: An International Journal, 77 (2017), 211-225.  doi: 10.1016/j.eswa.2017.01.042.  Google Scholar

[18]

F. Li, J. Song, A. Dolgui and L. Liang, Using common weights and efficiency invariance principles for resource allocation and target setting, International J Prod Res, 55 (2017), 4982-4997.  doi: 10.1080/00207543.2017.1287450.  Google Scholar

[19]

R. Lin, Z. Chen and Z. Li, A new approach for allocating fixed costs among decision making units, J Ind Manag Optim, 12 (2016), 211-228.  doi: 10.3934/jimo.2016.12.211.  Google Scholar

[20]

F. H. F. Liu and H. H. Peng, Ranking of units on the DEA frontier with common weights, Comput Oper Res, 35 (2008), 1624-1637.  doi: 10.1016/j.cor.2006.09.006.  Google Scholar

[21]

S. Mehrabian, G. R. Jahanshahloo, M. R. Alirezaei and G. R. Amin, An assurance interval of the non-archimedean epsilon in DEA models, Eur J Oper Res, 48 (2000), 189-350.  doi: 10.1287/opre.48.2.344.12381.  Google Scholar

[22]

J. X. Nan and D. F. Li, Linear programming technique for solving interval-valued constraint matrix games, J Ind Manag Optim Optimization, 10 (2014), 1059-1070.  doi: 10.3934/jimo.2014.10.1059.  Google Scholar

[23]

J. Pourmahmoud and Z. Zeynali, A nonlinear model for common weights set identification in network Data Envelopment Analysis, Int J Ind Math, 38 (2016), 87-98.   Google Scholar

[24]

A. Rahman, S. Lee and T. C. Chung, Accurate multi-criteria decision making methodology for recommending machine learning algorithm, J Expert Syst Appl: An International Journal archive, 71 (2017), 257-278.   Google Scholar

[25]

S. Ramazani-Tarkhorani, M. Khodabakhshi, S. Mehrabian and F. Nuri-Bahmani, Ranking decision-making units using common weights in DEA, Appl Math Model, 38 (2014), 3890-3896.  doi: 10.1016/j.apm.2013.08.029.  Google Scholar

[26]

Y. Roll, W. D. Cook and B. Golany, Controlling factor weighs in data envelopment analysis, IIE Trans, 23 (1991), 2-9.   Google Scholar

[27]

S. Saati, A. Hatami-Marbini, P. J. Agrell and M. Tavana, A common set of weight approach using an ideal decision making unit in data envelopment analysis, Journal of Industrial and Management Optimization, 8 (2012), 623-637.  doi: 10.3934/jimo.2012.8.623.  Google Scholar

[28]

M. Salahi, N. Torabi and A. Amiri, An optimistic robust optimization approach to common set of weights in DEA, Meas, 93 (2016), 67-73.  doi: 10.1016/j.measurement.2016.06.049.  Google Scholar

[29]

G. H. Shirdel, S. Ramezani-Tarkhorani and Z. Jafari, tA Method for Evaluating the Performance of Decision Making Units with Imprecise Data Using Common Set of Weights, Int J Appl Comput Math, 3 (2017), 411-423.  doi: 10.1007/s40819-016-0152-0.  Google Scholar

[30]

J. Sun, J. Wu and D. Guo, Performance ranking of units considering ideal and anti-ideal DMU with common weights, Appl Math Model, 37 (2013), 6301-6310.  doi: 10.1016/j.apm.2013.01.010.  Google Scholar

[31]

K. Tone, On returns to scale under weight restrictions in data envelopment analysis, J Prod Anal, 16 (2001), 31-47.   Google Scholar

[32]

Y. M. Wang, Y. Luo and Y. X. Lan, Common weights for fully ranking decision making units by regression analysis, Expert Syst Appl, 38 (2011), 9122-9128.  doi: 10.1016/j.eswa.2011.01.004.  Google Scholar

show all references

References:
[1]

N. Adler, L. Friedman and Z. Sinuany-Stern, Review of ranking methods in the data envelopment analysis context, Eur J Oper Res, 140 (2002), 249-265.  doi: 10.1016/S0377-2217(02)00068-1.  Google Scholar

[2]

N. Aghayi, M. Tavana and M. A. Raayatpanah, Robust efficiency measurement with common set of weights under varying degrees of conservatism and data uncertainty, Eur J. Ind Eng, 10 (2016), 385-405.  doi: 10.1504/EJIE.2016.076386.  Google Scholar

[3]

A. Aldamak and S. Zolfaghari, Review of efficiency ranking methods in data envelopment analysis, Meas, 106 (2017), 161-172.  doi: 10.1016/j.measurement.2017.04.028.  Google Scholar

[4]

R. D. Banker, A. Charnes and W. W. Cooper, Some models for estimating technical and scale inefficiencies in data envelopment analysis, Manage Sci, 30 (1984), 1031-1142.  doi: 10.1287/mnsc.30.9.1078.  Google Scholar

[5]

A. Charnes, W. W. Cooper and E. Rhodes, Measuring the efficiency of DMUs, Eur J Oper Res, 2 (1978), 429-444.  doi: 10.1016/0377-2217(78)90138-8.  Google Scholar

[6]

C. I. Chiang, M. J. Hwang and Y. H. Liu, Determining a common set of weights in a DEA problem using a separation vector, Math and Comput Model, 54 (2002), 2464-2470.  doi: 10.1016/j.mcm.2011.06.002.  Google Scholar

[7]

C. I. Chiang and G. H. Tzeng, A new efficiency measure for DEA: Efficiency achievement measure established on fuzzy multiple objectives programming, J Manage, 17 (2000), 369-388.   Google Scholar

[8]

W. D. Cook, Y. Roll and A. Kazakov, A DEA model for measuring the relative efficiency of highway maintenance patrols, Inform Syst Oper Res, 28 (1990), 113-124.   Google Scholar

[9]

D. K. Despotis, Improving the discriminating power of DEA: Focus on globally efficient units, J Oper Res Soc, 53 (2002), 314-323.  doi: 10.1057/palgrave.jors.2601253.  Google Scholar

[10]

J. A. Ganley and S. A. Cubbin, Public Sector Efficiency Measurement: Applications of Data Envelopment Analysis, North-Holland, Amsterdam, 1992. Google Scholar

[11]

A. Hatami Marbini, M. Tavana, P. J. Agrell, F. Hosseinzadeh Lotfi and Z. Ghelej Beigi, A common-weights DEA model for centralized resource reduction and target setting, Comput Ind Eng, 79 (2015), 195-203.   Google Scholar

[12]

F. Hosseinzadeh Lotfi, M. Rostamy-Malkhlifeh, G. R. Jahanshahloo, Z. Moghaddas, M. Khodabakhshi and M. Vaez-Ghasemi, A review of ranking models in data envelopment analysis, J Appl Math., 2013 (2013), Article ID: 492421, 20 pages. Google Scholar

[13]

C. K. Hu, F. B. Liu and C. F. Hu, Efficiency measures in fuzzy data envelopment analysis with common weights, J Ind Manage Opt, 13 (2017), 237-249.  doi: 10.3934/jimo.2016014.  Google Scholar

[14]

C. L. Hwang and A. S. M. Masud, Multiple Objective Decision Making: Methods and Applications, Springer, Berlin, 1979. Google Scholar

[15]

G. R. Jahanshahloo, F. Hosseinzadeh Lotfi, M. Khanmohammadi, M. Kazemimanesh and V. Rezaie, Ranking of units by positive ideal DMU with common weights, Expert Syst Appl, 37 (2010), 7483-7488.  doi: 10.1016/j.eswa.2010.04.011.  Google Scholar

[16]

C. Kao and H. Hung, Data envelopment analysis with common weights: The comprise solution approach, J Oper Res Soc, 56 (2005), 1196-1203.  doi: 10.1057/palgrave.jors.2601924.  Google Scholar

[17]

M. N. Kritikos, A full ranking methodology in data envelopment analysis based on a set of dummy decision making units, Expert Systems with Applications: An International Journal, 77 (2017), 211-225.  doi: 10.1016/j.eswa.2017.01.042.  Google Scholar

[18]

F. Li, J. Song, A. Dolgui and L. Liang, Using common weights and efficiency invariance principles for resource allocation and target setting, International J Prod Res, 55 (2017), 4982-4997.  doi: 10.1080/00207543.2017.1287450.  Google Scholar

[19]

R. Lin, Z. Chen and Z. Li, A new approach for allocating fixed costs among decision making units, J Ind Manag Optim, 12 (2016), 211-228.  doi: 10.3934/jimo.2016.12.211.  Google Scholar

[20]

F. H. F. Liu and H. H. Peng, Ranking of units on the DEA frontier with common weights, Comput Oper Res, 35 (2008), 1624-1637.  doi: 10.1016/j.cor.2006.09.006.  Google Scholar

[21]

S. Mehrabian, G. R. Jahanshahloo, M. R. Alirezaei and G. R. Amin, An assurance interval of the non-archimedean epsilon in DEA models, Eur J Oper Res, 48 (2000), 189-350.  doi: 10.1287/opre.48.2.344.12381.  Google Scholar

[22]

J. X. Nan and D. F. Li, Linear programming technique for solving interval-valued constraint matrix games, J Ind Manag Optim Optimization, 10 (2014), 1059-1070.  doi: 10.3934/jimo.2014.10.1059.  Google Scholar

[23]

J. Pourmahmoud and Z. Zeynali, A nonlinear model for common weights set identification in network Data Envelopment Analysis, Int J Ind Math, 38 (2016), 87-98.   Google Scholar

[24]

A. Rahman, S. Lee and T. C. Chung, Accurate multi-criteria decision making methodology for recommending machine learning algorithm, J Expert Syst Appl: An International Journal archive, 71 (2017), 257-278.   Google Scholar

[25]

S. Ramazani-Tarkhorani, M. Khodabakhshi, S. Mehrabian and F. Nuri-Bahmani, Ranking decision-making units using common weights in DEA, Appl Math Model, 38 (2014), 3890-3896.  doi: 10.1016/j.apm.2013.08.029.  Google Scholar

[26]

Y. Roll, W. D. Cook and B. Golany, Controlling factor weighs in data envelopment analysis, IIE Trans, 23 (1991), 2-9.   Google Scholar

[27]

S. Saati, A. Hatami-Marbini, P. J. Agrell and M. Tavana, A common set of weight approach using an ideal decision making unit in data envelopment analysis, Journal of Industrial and Management Optimization, 8 (2012), 623-637.  doi: 10.3934/jimo.2012.8.623.  Google Scholar

[28]

M. Salahi, N. Torabi and A. Amiri, An optimistic robust optimization approach to common set of weights in DEA, Meas, 93 (2016), 67-73.  doi: 10.1016/j.measurement.2016.06.049.  Google Scholar

[29]

G. H. Shirdel, S. Ramezani-Tarkhorani and Z. Jafari, tA Method for Evaluating the Performance of Decision Making Units with Imprecise Data Using Common Set of Weights, Int J Appl Comput Math, 3 (2017), 411-423.  doi: 10.1007/s40819-016-0152-0.  Google Scholar

[30]

J. Sun, J. Wu and D. Guo, Performance ranking of units considering ideal and anti-ideal DMU with common weights, Appl Math Model, 37 (2013), 6301-6310.  doi: 10.1016/j.apm.2013.01.010.  Google Scholar

[31]

K. Tone, On returns to scale under weight restrictions in data envelopment analysis, J Prod Anal, 16 (2001), 31-47.   Google Scholar

[32]

Y. M. Wang, Y. Luo and Y. X. Lan, Common weights for fully ranking decision making units by regression analysis, Expert Syst Appl, 38 (2011), 9122-9128.  doi: 10.1016/j.eswa.2011.01.004.  Google Scholar

Figure 1.  $0<y_P<x_P, \Delta_P = x_p-y_p ,m_p = {y_p \over x_p} , o<m_{p_1}<m_P<m_{p_2}$
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Figure 2.  The interior points of the area $OQ_1P$, $R_1$, have less amount of $\Delta$ than $P$ while slopes of crossing lines from the origin and these points are less than $m_P$
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Figure 3.  Slopes of crossing lines from the origin and the points located in the areas $R_1$ and $R_2$ are less than $m_P$
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Figure 4.  Range of slopes of the lines crossing the origin and the points located in $R_3$ is the same as that for the the points located in the area enclosed by the segments $OP$, $OQ_6$ and $Q_6P$.
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Figure 5.  Slope of the line crossing the origin and $P^{(U_1,V_1)}_o$ is maximum.
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Figure 6.  Two pairs of DMUs with the same efficiency scores.
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Table 1.  Input and Output data of DMUs
$DMU_j$ $x_{1j}$ $x_{2j}$ $x_{3j} $ $x_{4j}$ $y_{1j}$ $y_{2j} $
$DMU_1$99562051375262941271678
$DMU_2$91758981379204737211277
$DMU_3$3178100493615351127062051
$DMU_4$81358331124173021761538
$DMU_5$123686392486499052202042
$DMU_6 $114676101600358935171856
$DMU_7$70556001557362323522060
$DMU_8$2871115242880245217551664
$DMU_9$109889981730282344122334
$DMU_{10}$203293832421445453862080
$DMU_{11}$1414104682140364957352691
$DMU_{12}$1967112602759317860792804
$DMU_{13}$185198802335457058932495
$DMU_{14}$3100156495487294052483692
$DMU_{15}$5016180104008356778004852
$DMU_{16}$1924126822490297560403396
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$DMU_j$ $x_{1j}$ $x_{2j}$ $x_{3j} $ $x_{4j}$ $y_{1j}$ $y_{2j} $
$DMU_1$99562051375262941271678
$DMU_2$91758981379204737211277
$DMU_3$3178100493615351127062051
$DMU_4$81358331124173021761538
$DMU_5$123686392486499052202042
$DMU_6 $114676101600358935171856
$DMU_7$70556001557362323522060
$DMU_8$2871115242880245217551664
$DMU_9$109889981730282344122334
$DMU_{10}$203293832421445453862080
$DMU_{11}$1414104682140364957352691
$DMU_{12}$1967112602759317860792804
$DMU_{13}$185198802335457058932495
$DMU_{14}$3100156495487294052483692
$DMU_{15}$5016180104008356778004852
$DMU_{16}$1924126822490297560403396
Table 2.  The generated common set of weights
$v^*_1$ $v^*_2$ $v^*_3 $ $v^*_4$ $u^*_1$ $u^*_2 $
Our method0.0000010.1810810.0000010.1243920.1164660.578059
Liu and Peng's method [20]0.0000010.0000010.0000017.50E-078.70E- 070.000004
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$v^*_1$ $v^*_2$ $v^*_3 $ $v^*_4$ $u^*_1$ $u^*_2 $
Our method0.0000010.1810810.0000010.1243920.1164660.578059
Liu and Peng's method [20]0.0000010.0000010.0000017.50E-078.70E- 070.000004
Table 3.  The efficiencies
$DMU_j$ $E^*_j$CWA-EfficiencyCCR-Efficiency
$DMU_1$111
$DMU_2$0.8857610.8770071
$DMU_3$0.6651010.557160.690363
$DMU_4$0.898570.9114881
$DMU_5$0.8184360.8077231
$DMU_6$0.8125570.8239970.881091
$DMU_7$111
$DMU_8$0.487620.4404690.555791
$DMU_9$0.9406760.9689861
$DMU_{10}$0.8120470.7748540.863042
$DMU_{11}$0.946380.9632530.996068
$DMU_{12}$0.9566930.9205911
$DMU_{13}$0.902880.8842990.915511
$DMU_{14}$0.8580840.751161
$DMU_{15}$0.9600840.8674781
$DMU_{16}$111
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$DMU_j$ $E^*_j$CWA-EfficiencyCCR-Efficiency
$DMU_1$111
$DMU_2$0.8857610.8770071
$DMU_3$0.6651010.557160.690363
$DMU_4$0.898570.9114881
$DMU_5$0.8184360.8077231
$DMU_6$0.8125570.8239970.881091
$DMU_7$111
$DMU_8$0.487620.4404690.555791
$DMU_9$0.9406760.9689861
$DMU_{10}$0.8120470.7748540.863042
$DMU_{11}$0.946380.9632530.996068
$DMU_{12}$0.9566930.9205911
$DMU_{13}$0.902880.8842990.915511
$DMU_{14}$0.8580840.751161
$DMU_{15}$0.9600840.8674781
$DMU_{16}$111
Table 4.  The average of the efficiencies in each method and their ratio
Our methodLiu and Peng's method [20]The ratio
0.8715560.8467791.02926
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Our methodLiu and Peng's method [20]The ratio
0.8715560.8467791.02926
Table 5.  The ranking scores
$DMU_j$Our method: Input-oriented approachOur method: Output-oriented approachCWA-rankingCCR
$DMU_1$1321
$DMU_2$101091
$DMU_3$1515156
$DMU_4$9971
$DMU_5$1212121
$DMU_6$1313114
$DMU_7$2231
$DMU_8$1616167
$DMU_9$7741
$DMU_{10}$1414135
$DMU_{11}$6652
$DMU_{12}$5561
$DMU_{13}$8883
$DMU_{14}$1111141
$DMU_{15}$44101
$DMU_{16}$3111
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$DMU_j$Our method: Input-oriented approachOur method: Output-oriented approachCWA-rankingCCR
$DMU_1$1321
$DMU_2$101091
$DMU_3$1515156
$DMU_4$9971
$DMU_5$1212121
$DMU_6$1313114
$DMU_7$2231
$DMU_8$1616167
$DMU_9$7741
$DMU_{10}$1414135
$DMU_{11}$6652
$DMU_{12}$5561
$DMU_{13}$8883
$DMU_{14}$1111141
$DMU_{15}$44101
$DMU_{16}$3111
Table 6.  The averages of the obtained efficiencies and their ratio in each case
Size of the sampleOur methodCWA-EfficienciesThe ratio
60.9161280.9069611.010107
120.5390860.5390821.000006
250.6899370.1481764.656189
500.5507240.5533200.995309
1000.6392430.4716781.355254
2000.5823100.5807111.002753
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Size of the sampleOur methodCWA-EfficienciesThe ratio
60.9161280.9069611.010107
120.5390860.5390821.000006
250.6899370.1481764.656189
500.5507240.5533200.995309
1000.6392430.4716781.355254
2000.5823100.5807111.002753
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