DMU | A | B | C | D | Agregated |
input1 | 1 | 2 | 6 | 3 | 12 |
input2 | 5 | 2 | 1 | 3 | 11 |
output | 1 | 1 | 1 | 1 | 4 |
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: |
Table 1. 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 |
Table 2. Optimal solution of model (5) and virtual inputs and outputs connected with it for the example
Table 3. Data set of [14]
DMU | Input1 | Input2 | Input3 | Output1 | Output2 | |
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 |
Table 4. 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 |
Table 5. 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 |
[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. |
[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. |
[3] | R. D. Banker, Estimating most productive scale size using data envelopment analysis, European Journal of Operational Research, 17 (1984), 35-44. |
[4] | J. E. Beasley, Allocating fixed costs and resources via data envelopment analysis, European Journal of Operational Research, 147 (2003), 198-216. |
[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. |
[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. |
[7] | M. Carrillo and J. M. Jorge, A multiobjective dea approach to ranking alternatives, Expert Systems with Applications, 50 (2016), 130-139. |
[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. |
[9] | C. T. Chang, Fuzzy linearization strategy for multiple objective linear fractional programming with binary utility functions, Computers & Industrial Engineering, 112 (2017), 437-446. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[18] | J. P. Costa, Computing non-dominated solutions in molfp, European Journal of Operational Research, 181 (2007), 1464-1475. |
[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. |
[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. |
[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. |
[22] | W. Dinkelbach, On nonlinear fractional programming, Management Science, 13 (1967), 492-498. doi: 10.1287/mnsc.13.7.492. |
[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. |
[24] | M. Ehrgott, Multicriteria Optimization, 2$^{nd}$ edition, Springer, Berlin, 2005. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[31] | J. SH. Kornbluth and R. E. Steuer, Multiple objective linear fractional programming, Management Science, 27 (1981), 1024-1039. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[42] | S. Lozano and G. Villa, Centralized resource allocation using data envelopment analysis, Journal of Productivity Analysis, 22 (2004), 143-161. |
[43] | OB. Olesen, Some unsolved problems in data envelopment analysis: A survey, International Journal of Production Economics, 39 (1995), 5-36. |
[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. |
[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. |
[46] | Y. Roll, W. D. Cook and B. Golany, Controlling factor weights in data envelopment analysis, IIE transactions, 23 (1991), 2-9. |
[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. |
[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. |
[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. |
[50] | R. E. Steuer, Multiple Criteria Optimization, Wiley, New York, 1986. |
[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. |
[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. |
[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. |
[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. |