# American Institute of Mathematical Sciences

September  2020, 10(3): 257-273. doi: 10.3934/naco.2020001

## Resource allocation: A common set of weights model

 1 Department of Mathematics, Faculty of Mathematical Sciences and Computer, Kharazmi University, Tehran, Iran 2 Department of Mathematics, Islamic Azad University, Mashhad Branch, Mashhad, Iran

* Corresponding author: Sedighe Asghariniya

Received  November 2018 Revised  October 2019 Published  February 2020

Allocation problem is an important issue in management. Data envelopment analysis (DEA) is a non-parametric method for assessing a set of decision making units (DMUs). It has proven to be a useful technique to solve allocation problems. In recent years, many papers have been published in this regard and many researchers have tried to find a suitable allocation model based on DEA. Common set of weights (CSWs) is a DEA model which, in contrast with traditional DEA models, does not allow individual weights for each decision making unit. In this manner, all DMUs are assessed through choosing a same set of weights. In this article, we will use the weighted-sum method to solve the multi-objective CSW problem. Then, via introducing a set of special weights, we will connect the CSW model to a non-linear (fractional) CSW model. After linearization, the proposed model is used for allocating resources. To illustrate our model, some examples are also provided.

Citation: Sedighe Asghariniya, Hamed Zhiani Rezai, Saeid Mehrabian. Resource allocation: A common set of weights model. Numerical Algebra, Control & Optimization, 2020, 10 (3) : 257-273. doi: 10.3934/naco.2020001
##### References:
 [1] Y. Almogy and O. Levin, A class of fractional programming problems, Operations Research, 19 (1971), 57-67.  doi: 10.1287/opre.19.1.57.  Google Scholar [2] A. Amirteimoori and S. Kordrostami, Allocating fixed costs and target setting: A dea-based approach, Applied Mathematics and Computation, 171 (2005), 136-151.  doi: 10.1016/j.amc.2005.01.064.  Google Scholar [3] R. D. Banker, Estimating most productive scale size using data envelopment analysis, European Journal of Operational Research, 17 (1984), 35-44.   Google Scholar [4] J. E. Beasley, Allocating fixed costs and resources via data envelopment analysis, European Journal of Operational Research, 147 (2003), 198-216.   Google Scholar [5] G. Bi, J. Ding, Y. Luo and and L. Liang, Resource allocation and target setting for parallel production system based on dea, Applied Mathematical Modelling, 35 (2011), 4270-4280.  doi: 10.1016/j.apm.2011.02.048.  Google Scholar [6] A. Biswas and A.K. De, A priority based fuzzy programming approach for multiobjective probabilistic linear fractional programming, In 2013 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), (2013), 1–6. Google Scholar [7] M. Carrillo and J. M. Jorge, A multiobjective dea approach to ranking alternatives, Expert Systems with Applications, 50 (2016), 130-139.   Google Scholar [8] M. Chakraborty and S. Gupta, Fuzzy mathematical programming for multi objective linear fractional programming problem, Fuzzy Sets and Systems, 125 (2002), 335-342.  doi: 10.1016/S0165-0114(01)00060-4.  Google Scholar [9] C. T. Chang, Fuzzy linearization strategy for multiple objective linear fractional programming with binary utility functions, Computers & Industrial Engineering, 112 (2017), 437-446.   Google Scholar [10] A. Charnes and W. W. Cooper, Programming with linear fractional functionals, Naval Research Logistics Quarterly, 9 (1962), 181-186.  doi: 10.1002/nav.3800090303.  Google Scholar [11] A. Charnes, W. W. Cooper and E. Rhodes, Measuring the efficiency of decision making units, European Journal of Operational Research, 2 (1978), 429-444.  doi: 10.1016/0377-2217(78)90138-8.  Google Scholar [12] C. Chiang, M. J. Hwang and Y. H. Liu, etermining a common set of weights in a dea problem using a separation vector, Mathematical and Computer Modelling, 54 (2011), 2464-2470.  doi: 10.1016/j.mcm.2011.06.002.  Google Scholar [13] C. I. Chiang and G. H. Tzeng, A new efficiency measure for dea: efficiency achievement measure established on fuzzy multiple objectives programming, Journal of Management, 17 (2000), 369-388.   Google Scholar [14] W. D. Cook and M. Kress, Characterizing an equitable allocation of shared costs: A dea approach, European Journal of Operational Research, 119 (1999), 652-661.   Google Scholar [15] W. D. Cook, Y. Roll and A. Kazakov, A dea model for measuring the relative eeficiency of highway maintenance patrols, INFOR: Information Systems and Operational Research, 28 (1990), 113-124.   Google Scholar [16] W. D. Cook and J. Zhu, Allocation of shared costs among decision making units: A dea approach, Computers & Operations Research, 32 (2005), 2171-2178.   Google Scholar [17] W. W. Cooper, L. M. Seiford and K. Tone, Data Envelopment Analysis: A Comprehensive Text with Models, Applications, References and DEA-Solver Software, 2$^{nd}$ edition, Springer, New York, 2006. Google Scholar [18] J. P. Costa, Computing non-dominated solutions in molfp, European Journal of Operational Research, 181 (2007), 1464-1475.   Google Scholar [19] J. P. Costa and M. J. Alves, A reference point technique to compute nondominated solutions in molfp, Journal of Mathematical Sciences, 161 (2009), 820-831.  doi: 10.1007/s10958-009-9603-z.  Google Scholar [20] Y. Dai, J. Shi and S. Wang, Conical partition algorithm for maximizing the sum of dc ratios, Journal of Global Optimization, 31 (2005), 253-270.  doi: 10.1007/s10898-004-5699-3.  Google Scholar [21] P. K. De and M. Deb, Solution of multi objective linear fractional programming problem by taylor series approach, In 2015 International Conference on Man and Machine Interfacing (MAMI), pages 1–5. IEEE, 2015. Google Scholar [22] W. Dinkelbach, On nonlinear fractional programming, Management Science, 13 (1967), 492-498.  doi: 10.1287/mnsc.13.7.492.  Google Scholar [23] J. Du, W. D. Cook, L. Liang and J. Zhu, Fixed cost and resource allocation based on dea cross-efficiency, European Journal of Operational Research, 235 (2014), 206-214.  doi: 10.1016/j.ejor.2013.10.002.  Google Scholar [24] M. Ehrgott, Multicriteria Optimization, 2$^{nd}$ edition, Springer, Berlin, 2005.  Google Scholar [25] J. E. Falk and S. W. Palocsay, Image space analysis of generalized fractional programs, Journal of Global Optimization, 4 (1994), 63-88.  doi: 10.1007/BF01096535.  Google Scholar [26] N. Güzel, A proposal to the solution of multiobjective linear fractional programming problem, Abstract and Applied Analysis, 2013 (2013), Article ID 435030, 4 pages. doi: 10.1155/2013/435030.  Google Scholar [27] G. R. Jahanshahloo, A. Memariani, F. Hosseinzadeh Lotfi 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), 265-281.  doi: 10.1016/j.amc.2004.04.088.  Google Scholar [28] G. R. Jahanshahloo, J. Sadeghi and M. Khodabakhshi, Proposing a method for fixed cost allocation using dea based on the efficiency invariance and common set of weights principles, Mathematical Methods of Operations Research, 85 (2017), 223-240.  doi: 10.1007/s00186-016-0563-z.  Google Scholar [29] G. R. Jahanshahloo, B. Talebian, F. Hosseinzadeh Lotfi and J. Sadeghi, Finding a solution for multi-objective linear fractional programming problem based on goal programming and data envelopment analysis, RAIRO-Operations Research, 51 (2017), 199-210.  doi: 10.1051/ro/2016014.  Google Scholar [30] 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.   Google Scholar [31] J. SH. Kornbluth and R. E. Steuer, Multiple objective linear fractional programming, Management Science, 27 (1981), 1024-1039.   Google Scholar [32] T. Kuno, A branch-and-bound algorithm for maximizing the sum of several linear ratios, Journal of Global Optimization, 22 (2002), 155-174.  doi: 10.1023/A:1013807129844.  Google Scholar [33] F. Li, J. Song, A. Dolgui and L. Liang, Using common weights and efficiency invariance principles for resource allocation and target setting, International Journal of Production Research, 55 (2017), 4982-4997.   Google Scholar [34] F. Li, Q. Zhu and L. Liang, Allocating a fixed cost based on a dea-game cross efficiency approach, Expert Systems with Applications, 96 (2018), 196-207.  doi: 10.1007/s11424-015-4211-0.  Google Scholar [35] Y. Li, F. Yang, L. Liang and Z. Hua, Allocating the fixed cost as a complement of other cost inputs: A dea approach, European Journal of Operational Research, 197 (2009), 389-401.  doi: 10.1016/j.ejor.2008.06.017.  Google Scholar [36] Y. Li, M. Yang, Y. Chen, Q. Dai and L. Liang, Allocating a fixed cost based on data envelopment analysis and satisfaction degree, Omega, 41 (2013), 55-60.   Google Scholar [37] R. Y. Lin., Allocating fixed costs or resources and setting targets via data envelopment analysis, Applied Mathematics and Computation, 217 (2011), 6349-6358.  doi: 10.1016/j.amc.2011.01.008.  Google Scholar [38] R. Y. Lin and Z. Chen, Fixed input allocation methods based on super ccr efficiency invariance and practical feasibility, Applied Mathematical Modelling, 40 (2016), 5377-5392.  doi: 10.1016/j.apm.2015.06.039.  Google Scholar [39] F. H. F. Liu and H. H. Peng, Ranking of units on the dea frontier with common weights, Computers & Operations Research, 35 (2008), 1624-1637.   Google Scholar [40] F. Hosseinzadeh Lotfi, A. Hatami-Marbini, P.J. Agrell, N. Aghayi and K. Gholami, Allocating fixed resources and setting targets using a common-weights dea approach, Computers & Industrial Engineering, 64 (2013), 631-640.   Google Scholar [41] F. Hosseinzadeh Lotfi, A. A. Noora, G. R. Jahanshahloo, M. Khodabakhshi and A. Payan, A linear programming approach to test efficiency in multi-objective linear fractional programming problems, Applied Mathematical Modelling, 34 (2010), 4179-4183.  doi: 10.1016/j.apm.2010.04.015.  Google Scholar [42] S. Lozano and G. Villa, Centralized resource allocation using data envelopment analysis, Journal of Productivity Analysis, 22 (2004), 143-161.   Google Scholar [43] OB. Olesen, Some unsolved problems in data envelopment analysis: A survey, International Journal of Production Economics, 39 (1995), 5-36.   Google Scholar [44] S. Ramezani-Tarkhorani, M. Khodabakhshi, S. Mehrabian and F. Nuri-Bahmani, Ranking decision-making units using common weights in dea, Applied Mathematical Modelling, 38 (2014), 3890-3896.  doi: 10.1016/j.apm.2013.08.029.  Google Scholar [45] N. Ramón, J. Ruiz and I. Sirvent, Common sets of weights as summaries of dea profiles of weights: With an application to the ranking of professional tennis players, Expert Systems with Applications, 39 (2012), 4882-4889.   Google Scholar [46] Y. Roll, W. D. Cook and B. Golany, Controlling factor weights in data envelopment analysis, IIE transactions, 23 (1991), 2-9.   Google Scholar [47] 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 [48] S. Sadia, N. Gupta, Q. M. Ali and A. Bari, Solving multi-objective linear plus linear fractional programming problem, Journal of Statistics Applications & Probability, 4 (2015), 253-258.   Google Scholar [49] X. Si, L. Liang, G. Jia, L. Yang, H. Wu and Y. Li, Proportional sharing and dea in allocating the fixed cost, Applied Mathematics and Computation, 219 (2013), 6580-6590.  doi: 10.1016/j.amc.2012.12.085.  Google Scholar [50] R. E. Steuer, Multiple Criteria Optimization, Wiley, New York, 1986.  Google Scholar [51] E. Valipour, M. A. Yaghoobi and M. Mashinchi, An iterative approach to solve multiobjective linear fractional programming problems, Applied Mathematical Modelling, 38 (2014), 38-49.  doi: 10.1016/j.apm.2013.05.046.  Google Scholar [52] H. Yan, Q. Wei and G. Hao, Dea models for resource reallocation and production input/output estimation, European Journal of Operational Research, 136 (2002), 19-31.  doi: 10.1016/S0377-2217(01)00046-7.  Google Scholar [53] Z. Yang and Q. Zhang, Resource allocation based on dea and modified shapley value, Applied Mathematics and Computation, 263 (2015), 280-286.  doi: 10.1016/j.amc.2015.04.063.  Google Scholar [54] M. Zohrehbandian, A. Makui and A. Alinezhad, A compromise solution approach for finding common weights in dea: An improvement to kao and hung's approach, Journal of the Operational Research Society, 61 (2010), 604-610.   Google Scholar

show all references

##### References:
 [1] Y. Almogy and O. Levin, A class of fractional programming problems, Operations Research, 19 (1971), 57-67.  doi: 10.1287/opre.19.1.57.  Google Scholar [2] A. Amirteimoori and S. Kordrostami, Allocating fixed costs and target setting: A dea-based approach, Applied Mathematics and Computation, 171 (2005), 136-151.  doi: 10.1016/j.amc.2005.01.064.  Google Scholar [3] R. D. Banker, Estimating most productive scale size using data envelopment analysis, European Journal of Operational Research, 17 (1984), 35-44.   Google Scholar [4] J. E. Beasley, Allocating fixed costs and resources via data envelopment analysis, European Journal of Operational Research, 147 (2003), 198-216.   Google Scholar [5] G. Bi, J. Ding, Y. Luo and and L. Liang, Resource allocation and target setting for parallel production system based on dea, Applied Mathematical Modelling, 35 (2011), 4270-4280.  doi: 10.1016/j.apm.2011.02.048.  Google Scholar [6] A. Biswas and A.K. De, A priority based fuzzy programming approach for multiobjective probabilistic linear fractional programming, In 2013 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), (2013), 1–6. Google Scholar [7] M. Carrillo and J. M. Jorge, A multiobjective dea approach to ranking alternatives, Expert Systems with Applications, 50 (2016), 130-139.   Google Scholar [8] M. Chakraborty and S. Gupta, Fuzzy mathematical programming for multi objective linear fractional programming problem, Fuzzy Sets and Systems, 125 (2002), 335-342.  doi: 10.1016/S0165-0114(01)00060-4.  Google Scholar [9] C. T. Chang, Fuzzy linearization strategy for multiple objective linear fractional programming with binary utility functions, Computers & Industrial Engineering, 112 (2017), 437-446.   Google Scholar [10] A. Charnes and W. W. Cooper, Programming with linear fractional functionals, Naval Research Logistics Quarterly, 9 (1962), 181-186.  doi: 10.1002/nav.3800090303.  Google Scholar [11] A. Charnes, W. W. Cooper and E. Rhodes, Measuring the efficiency of decision making units, European Journal of Operational Research, 2 (1978), 429-444.  doi: 10.1016/0377-2217(78)90138-8.  Google Scholar [12] C. Chiang, M. J. Hwang and Y. H. Liu, etermining a common set of weights in a dea problem using a separation vector, Mathematical and Computer Modelling, 54 (2011), 2464-2470.  doi: 10.1016/j.mcm.2011.06.002.  Google Scholar [13] C. I. Chiang and G. H. Tzeng, A new efficiency measure for dea: efficiency achievement measure established on fuzzy multiple objectives programming, Journal of Management, 17 (2000), 369-388.   Google Scholar [14] W. D. Cook and M. Kress, Characterizing an equitable allocation of shared costs: A dea approach, European Journal of Operational Research, 119 (1999), 652-661.   Google Scholar [15] W. D. Cook, Y. Roll and A. Kazakov, A dea model for measuring the relative eeficiency of highway maintenance patrols, INFOR: Information Systems and Operational Research, 28 (1990), 113-124.   Google Scholar [16] W. D. Cook and J. Zhu, Allocation of shared costs among decision making units: A dea approach, Computers & Operations Research, 32 (2005), 2171-2178.   Google Scholar [17] W. W. Cooper, L. M. Seiford and K. Tone, Data Envelopment Analysis: A Comprehensive Text with Models, Applications, References and DEA-Solver Software, 2$^{nd}$ edition, Springer, New York, 2006. Google Scholar [18] J. P. Costa, Computing non-dominated solutions in molfp, European Journal of Operational Research, 181 (2007), 1464-1475.   Google Scholar [19] J. P. Costa and M. J. Alves, A reference point technique to compute nondominated solutions in molfp, Journal of Mathematical Sciences, 161 (2009), 820-831.  doi: 10.1007/s10958-009-9603-z.  Google Scholar [20] Y. Dai, J. Shi and S. Wang, Conical partition algorithm for maximizing the sum of dc ratios, Journal of Global Optimization, 31 (2005), 253-270.  doi: 10.1007/s10898-004-5699-3.  Google Scholar [21] P. K. De and M. Deb, Solution of multi objective linear fractional programming problem by taylor series approach, In 2015 International Conference on Man and Machine Interfacing (MAMI), pages 1–5. IEEE, 2015. Google Scholar [22] W. Dinkelbach, On nonlinear fractional programming, Management Science, 13 (1967), 492-498.  doi: 10.1287/mnsc.13.7.492.  Google Scholar [23] J. Du, W. D. Cook, L. Liang and J. Zhu, Fixed cost and resource allocation based on dea cross-efficiency, European Journal of Operational Research, 235 (2014), 206-214.  doi: 10.1016/j.ejor.2013.10.002.  Google Scholar [24] M. Ehrgott, Multicriteria Optimization, 2$^{nd}$ edition, Springer, Berlin, 2005.  Google Scholar [25] J. E. Falk and S. W. Palocsay, Image space analysis of generalized fractional programs, Journal of Global Optimization, 4 (1994), 63-88.  doi: 10.1007/BF01096535.  Google Scholar [26] N. Güzel, A proposal to the solution of multiobjective linear fractional programming problem, Abstract and Applied Analysis, 2013 (2013), Article ID 435030, 4 pages. doi: 10.1155/2013/435030.  Google Scholar [27] G. R. Jahanshahloo, A. Memariani, F. Hosseinzadeh Lotfi 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), 265-281.  doi: 10.1016/j.amc.2004.04.088.  Google Scholar [28] G. R. Jahanshahloo, J. Sadeghi and M. Khodabakhshi, Proposing a method for fixed cost allocation using dea based on the efficiency invariance and common set of weights principles, Mathematical Methods of Operations Research, 85 (2017), 223-240.  doi: 10.1007/s00186-016-0563-z.  Google Scholar [29] G. R. Jahanshahloo, B. Talebian, F. Hosseinzadeh Lotfi and J. Sadeghi, Finding a solution for multi-objective linear fractional programming problem based on goal programming and data envelopment analysis, RAIRO-Operations Research, 51 (2017), 199-210.  doi: 10.1051/ro/2016014.  Google Scholar [30] 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.   Google Scholar [31] J. SH. Kornbluth and R. E. Steuer, Multiple objective linear fractional programming, Management Science, 27 (1981), 1024-1039.   Google Scholar [32] T. Kuno, A branch-and-bound algorithm for maximizing the sum of several linear ratios, Journal of Global Optimization, 22 (2002), 155-174.  doi: 10.1023/A:1013807129844.  Google Scholar [33] F. Li, J. Song, A. Dolgui and L. Liang, Using common weights and efficiency invariance principles for resource allocation and target setting, International Journal of Production Research, 55 (2017), 4982-4997.   Google Scholar [34] F. Li, Q. Zhu and L. Liang, Allocating a fixed cost based on a dea-game cross efficiency approach, Expert Systems with Applications, 96 (2018), 196-207.  doi: 10.1007/s11424-015-4211-0.  Google Scholar [35] Y. Li, F. Yang, L. Liang and Z. Hua, Allocating the fixed cost as a complement of other cost inputs: A dea approach, European Journal of Operational Research, 197 (2009), 389-401.  doi: 10.1016/j.ejor.2008.06.017.  Google Scholar [36] Y. Li, M. Yang, Y. Chen, Q. Dai and L. Liang, Allocating a fixed cost based on data envelopment analysis and satisfaction degree, Omega, 41 (2013), 55-60.   Google Scholar [37] R. Y. Lin., Allocating fixed costs or resources and setting targets via data envelopment analysis, Applied Mathematics and Computation, 217 (2011), 6349-6358.  doi: 10.1016/j.amc.2011.01.008.  Google Scholar [38] R. Y. Lin and Z. Chen, Fixed input allocation methods based on super ccr efficiency invariance and practical feasibility, Applied Mathematical Modelling, 40 (2016), 5377-5392.  doi: 10.1016/j.apm.2015.06.039.  Google Scholar [39] F. H. F. Liu and H. H. Peng, Ranking of units on the dea frontier with common weights, Computers & Operations Research, 35 (2008), 1624-1637.   Google Scholar [40] F. Hosseinzadeh Lotfi, A. Hatami-Marbini, P.J. Agrell, N. Aghayi and K. Gholami, Allocating fixed resources and setting targets using a common-weights dea approach, Computers & Industrial Engineering, 64 (2013), 631-640.   Google Scholar [41] F. Hosseinzadeh Lotfi, A. A. Noora, G. R. Jahanshahloo, M. Khodabakhshi and A. Payan, A linear programming approach to test efficiency in multi-objective linear fractional programming problems, Applied Mathematical Modelling, 34 (2010), 4179-4183.  doi: 10.1016/j.apm.2010.04.015.  Google Scholar [42] S. Lozano and G. Villa, Centralized resource allocation using data envelopment analysis, Journal of Productivity Analysis, 22 (2004), 143-161.   Google Scholar [43] OB. Olesen, Some unsolved problems in data envelopment analysis: A survey, International Journal of Production Economics, 39 (1995), 5-36.   Google Scholar [44] S. Ramezani-Tarkhorani, M. Khodabakhshi, S. Mehrabian and F. Nuri-Bahmani, Ranking decision-making units using common weights in dea, Applied Mathematical Modelling, 38 (2014), 3890-3896.  doi: 10.1016/j.apm.2013.08.029.  Google Scholar [45] N. Ramón, J. Ruiz and I. Sirvent, Common sets of weights as summaries of dea profiles of weights: With an application to the ranking of professional tennis players, Expert Systems with Applications, 39 (2012), 4882-4889.   Google Scholar [46] Y. Roll, W. D. Cook and B. Golany, Controlling factor weights in data envelopment analysis, IIE transactions, 23 (1991), 2-9.   Google Scholar [47] 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 [48] S. Sadia, N. Gupta, Q. M. Ali and A. Bari, Solving multi-objective linear plus linear fractional programming problem, Journal of Statistics Applications & Probability, 4 (2015), 253-258.   Google Scholar [49] X. Si, L. Liang, G. Jia, L. Yang, H. Wu and Y. Li, Proportional sharing and dea in allocating the fixed cost, Applied Mathematics and Computation, 219 (2013), 6580-6590.  doi: 10.1016/j.amc.2012.12.085.  Google Scholar [50] R. E. Steuer, Multiple Criteria Optimization, Wiley, New York, 1986.  Google Scholar [51] E. Valipour, M. A. Yaghoobi and M. Mashinchi, An iterative approach to solve multiobjective linear fractional programming problems, Applied Mathematical Modelling, 38 (2014), 38-49.  doi: 10.1016/j.apm.2013.05.046.  Google Scholar [52] H. Yan, Q. Wei and G. Hao, Dea models for resource reallocation and production input/output estimation, European Journal of Operational Research, 136 (2002), 19-31.  doi: 10.1016/S0377-2217(01)00046-7.  Google Scholar [53] Z. Yang and Q. Zhang, Resource allocation based on dea and modified shapley value, Applied Mathematics and Computation, 263 (2015), 280-286.  doi: 10.1016/j.amc.2015.04.063.  Google Scholar [54] M. Zohrehbandian, A. Makui and A. Alinezhad, A compromise solution approach for finding common weights in dea: An improvement to kao and hung's approach, Journal of the Operational Research Society, 61 (2010), 604-610.   Google Scholar
The graph of the weighted-sum example
PPS in a two dimensions case
PPS$_{CCR}$ for the peresentedDMUs in Table 1
virtual plane of the example
Information related to example
 DMU A B C D Agregated input1 1 2 6 3 12 input2 5 2 1 3 11 output 1 1 1 1 4
 DMU A B C D Agregated input1 1 2 6 3 12 input2 5 2 1 3 11 output 1 1 1 1 4
Optimal solution of model (5) and virtual inputs and outputs connected with it for the example
 $u^*$ $v^*$ $({v^ * }{x_1},{u^ * }{y_1})$ $({v^ * }{x_2},{u^ * }{y_2})$ $({v^ * }{x_3},{u^ * }{y_3})$ $({v^ * }{x_4},{u^ * }{y_4})$ $({v^ * }{\bar x},{u^ * }{\bar y})$ $\frac{8}{47}$ $(\frac{8}{47},\frac{1}{47})$ $(\frac{8}{47},\frac{8}{47})$ $(\frac{8}{47},\frac{8}{47})$ $(\frac{10}{47},\frac{8}{47})$ $(\frac{12}{47},\frac{8}{47})$ $(\frac{32}{47},\frac{47}{47})$
 $u^*$ $v^*$ $({v^ * }{x_1},{u^ * }{y_1})$ $({v^ * }{x_2},{u^ * }{y_2})$ $({v^ * }{x_3},{u^ * }{y_3})$ $({v^ * }{x_4},{u^ * }{y_4})$ $({v^ * }{\bar x},{u^ * }{\bar y})$ $\frac{8}{47}$ $(\frac{8}{47},\frac{1}{47})$ $(\frac{8}{47},\frac{8}{47})$ $(\frac{8}{47},\frac{8}{47})$ $(\frac{10}{47},\frac{8}{47})$ $(\frac{12}{47},\frac{8}{47})$ $(\frac{32}{47},\frac{47}{47})$
Data set of [14]
 DMU Input1 Input2 Input3 Output1 Output2 $EFF_CCR$ 1 350 39 9 67 751 0.75663 2 298 26 8 73 611 0.92300 3 422 31 7 75 584 0.74384 4 281 16 9 70 665 1.00000 5 301 16 6 75 445 1.00000 6 360 29 17 83 1070 0.96112 7 540 18 10 72 457 0.85863 8 276 33 5 78 590 1.00000 9 323 25 5 75 1074 1.00000 10 444 64 6 74 1072 0.83102 11 323 25 5 25 350 0.33325 12 444 64 6 104 1199 1.00000
 DMU Input1 Input2 Input3 Output1 Output2 $EFF_CCR$ 1 350 39 9 67 751 0.75663 2 298 26 8 73 611 0.92300 3 422 31 7 75 584 0.74384 4 281 16 9 70 665 1.00000 5 301 16 6 75 445 1.00000 6 360 29 17 83 1070 0.96112 7 540 18 10 72 457 0.85863 8 276 33 5 78 590 1.00000 9 323 25 5 75 1074 1.00000 10 444 64 6 74 1072 0.83102 11 323 25 5 25 350 0.33325 12 444 64 6 104 1199 1.00000
Allocated cost to DMUs obtained by different methods
 DMU Allocated resource Efficiency DMU Allocated resource Efficiency 1 8.47412 1.00000 7 4.84712 1.00000 2 6.90521 1.00000 8 6.68270 1.00000 3 6.45563 1.00000 9 12.32408 1.00000 4 7.56372 1.00000 10 12.13035 1.00000 5 4.96678 1.00000 11 3.76675 1.00000 6 12.22981 1.00000 12 13.65374 1.00000
 DMU Allocated resource Efficiency DMU Allocated resource Efficiency 1 8.47412 1.00000 7 4.84712 1.00000 2 6.90521 1.00000 8 6.68270 1.00000 3 6.45563 1.00000 9 12.32408 1.00000 4 7.56372 1.00000 10 12.13035 1.00000 5 4.96678 1.00000 11 3.76675 1.00000 6 12.22981 1.00000 12 13.65374 1.00000
Different allocation in selected methods
 Eff. invariance Output orientation Input orientation no no no no no no yes no yes our Besley Du et al. Li et al. Hossein Zadeh Si et al. Cook and Lin and Yang and DMU approach [4] [23] [34] Lotfi et al. [40] [49] Kress [14] Chen [38] Zhang [53] 1 8.47 6.78 5.79 5.54 8.20 7.65 14.52 9.83 7.54 2 6.91 7.21 7.95 7.53 7.46 8.41 6.74 7.53 8.65 3 6.46 6.83 6.54 7.35 4.28 8.62 9.32 9.93 7.52 4 7.56 8.47 11.10 7.87 9.30 8.11 5.6 5.20 9.05 5 4.97 7.08 8.69 6.38 4.81 8.69 5.79 5.20 9.07 6 12.23 10.06 13.49 11.50 15.37 9.57 8.15 9.10 8.81 7 4.85 5.09 7.10 5.90 0 8.33 8.86 5.85 8.17 8 6.68 7.74 6.83 7.77 7.34 9.96 6.26 8.96 9.06 9 12.32 15.11 16.68 11.90 16.33 8.65 7.31 8.07 10.46 10 12.13 10.08 5.42 11.38 11.60 8.35 10.08 9.96 8.01 11 3.77 1.58 0 2.74 0 2.80 7.31 8.07 4.55 12 13.65 13.97 10.41 14.14 15.31 11.85 10.08 12.56 9.12 Gap 9.88 13.53 16.68 11.40 16.33 9.05 8.92 7.36 5.91
 Eff. invariance Output orientation Input orientation no no no no no no yes no yes our Besley Du et al. Li et al. Hossein Zadeh Si et al. Cook and Lin and Yang and DMU approach [4] [23] [34] Lotfi et al. [40] [49] Kress [14] Chen [38] Zhang [53] 1 8.47 6.78 5.79 5.54 8.20 7.65 14.52 9.83 7.54 2 6.91 7.21 7.95 7.53 7.46 8.41 6.74 7.53 8.65 3 6.46 6.83 6.54 7.35 4.28 8.62 9.32 9.93 7.52 4 7.56 8.47 11.10 7.87 9.30 8.11 5.6 5.20 9.05 5 4.97 7.08 8.69 6.38 4.81 8.69 5.79 5.20 9.07 6 12.23 10.06 13.49 11.50 15.37 9.57 8.15 9.10 8.81 7 4.85 5.09 7.10 5.90 0 8.33 8.86 5.85 8.17 8 6.68 7.74 6.83 7.77 7.34 9.96 6.26 8.96 9.06 9 12.32 15.11 16.68 11.90 16.33 8.65 7.31 8.07 10.46 10 12.13 10.08 5.42 11.38 11.60 8.35 10.08 9.96 8.01 11 3.77 1.58 0 2.74 0 2.80 7.31 8.07 4.55 12 13.65 13.97 10.41 14.14 15.31 11.85 10.08 12.56 9.12 Gap 9.88 13.53 16.68 11.40 16.33 9.05 8.92 7.36 5.91