# American Institute of Mathematical Sciences

2016, 12(1): 211-228. doi: 10.3934/jimo.2016.12.211

## A new approach for allocating fixed costs among decision making units

 1 Department of Computing Science, School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, Shaanxi, 710049, China, China, China

Received  March 2014 Revised  January 2015 Published  April 2015

How to equitably distribute a common fixed cost among decision making units (DMUs) of an organization is a typical problem in organization management. Based on the data envelopment analysis technique, this paper proposes a new proportional sharing model to determine a unique fixed cost allocation under two assumptions: efficiency invariance and zero slack. It is noteworthy that the fixed cost allocation determined by our proportional sharing model is a feasible solution to the model proposed by Cook and Zhu [Cook and Zhu, Allocation of shared costs among decision making units: A DEA approach, Computers & Operations Research, 32 (2005) 2171-2178]. To ensure the uniqueness of the fixed cost allocation, three algorithms are proposed under the new model. Different from current fixed cost allocation methods under the efficiency invariance assumption, our approach can generate a fixed cost allocation that is unique, partially dependent of DMUs' inputs and units-invariant, and thus is more effective. Numerical examples are used to illustrate the validity and superiorities of our approach.
Citation: Ruiyue Lin, Zhiping Chen, Zongxin Li. A new approach for allocating fixed costs among decision making units. Journal of Industrial & Management Optimization, 2016, 12 (1) : 211-228. doi: 10.3934/jimo.2016.12.211
##### References:
 [1] A. Amirteimoori and S. Kordrostami, Allocating fixed costs and target setting: A DEA-based approach,, Applied Mathematics and Computation, 171 (2005), 136. doi: 10.1016/j.amc.2005.01.064. [2] J. E. Beasley, Allocating fixed costs and resources via data envelopment analysis,, European Journal of Operational Research, 147 (2003), 198. doi: 10.1016/S0377-2217(02)00244-8. [3] A. Cadena, A. Marcucci, J. F. Pérez, H. Durán, H. Mutis, C. Taútiva and F. Palacios, Efficiency analysis in electricity transmission utilities,, Journal of Industrial and management optimization, 5 (2009), 253. doi: 10.3934/jimo.2009.5.253. [4] A. Charnes, W. W. Cooper and E. Rhodes, Measuring the efficiency of decision making units,, European Journal of Operational Research, 2 (1978), 429. doi: 10.1016/0377-2217(78)90138-8. [5] W. D. Cook and M. Kress, Characterizing an equitable allocation of shared costs: A DEA approach,, European Journal of Operational Research, 119 (1999), 652. doi: 10.1016/S0377-2217(98)00337-3. [6] W. D. Cook and J. Zhu, Allocation of shared costs among decision making units: A DEA approach,, Computers & Operations Research, 32 (2005), 2171. doi: 10.1016/j.cor.2004.02.007. [7] W. W. Cooper, L. M. Seiford and K. Tone, Data Envelopment Analysis,, 2nd edition, (2007). [8] 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. [9] G. R. Jahanshahloo, F. Hosseinzadeh Lotfi, N. Shoja and M. Sanei, An alternative approach for equitable allocation of shared costs by using DEA,, Applied Mathematics and Computation, 153 (2004), 267. doi: 10.1016/S0096-3003(03)00631-3. [10] M. Khodabakhshi and K. Aryavash, The fair allocation of common fixed cost or revenue using DEA concept,, Annals of Operational Research, 214 (2014), 187. doi: 10.1007/s10479-012-1117-2. [11] 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. doi: 10.1016/j.ejor.2008.06.017. [12] 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. doi: 10.1016/j.omega.2011.02.008. [13] R. Lin, Allocating fixed costs or resources and setting targets via data envelopment analysis,, Applied Mathematics and Computation, 217 (2011), 6349. doi: 10.1016/j.amc.2011.01.008. [14] R. Lin, Allocating fixed costs and common revenue via data envelopment analysis,, Applied Mathematics and Computation, 218 (2011), 3680. doi: 10.1016/j.amc.2011.09.011. [15] M. Mahdiloo, A. Noorizadeh and R. Farzipoor Saen, Developing a new data envelopment analysis model for custer value analysis,, Journal of Industrial and management optimization, 7 (2011), 531. [16] A. Z. Milioni, J. V. G. Avellar, E. G. Gomes and J. C. B. Soares de Mello, An ellipsoidal frontier model: Allocating input via parametric DEA,, European Journal of Operational Research, 209 (2011), 113. doi: 10.1016/j.ejor.2010.08.008. [17] A. Z. Milioni, E. C. C. Guedes, J. V. G. Avellar and R. C. Silva, Adjusted spherical frontier model: Allocating input via parametric DEA,, Journal of the Operational Research Society, 63 (2012), 406. [18] H. Moulin and R. Stong, Fair queuing and other probabilistic allocation methods,, Mathematics of Operations Research, 27 (2002), 1. doi: 10.1287/moor.27.1.1.336. [19] 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. doi: 10.1016/j.amc.2012.12.085. [20] R. C. Silva and A. Z. Milioni, The adjusted spherical drontier model with weight restrictions,, European Journal of Operational Research, 220 (2012), 729. doi: 10.1016/j.ejor.2012.01.064. [21] T. Sueyoshi and K. Sekitani, An occurrence of multiple projections in DEA-based measurement of technical efficiency: Theoretical comparison among DEA models from desirable properties,, European Journal of Operational Research, 196 (2009), 764. doi: 10.1016/j.ejor.2008.01.045. [22] Y. T. Wang and D. X. Zhu, Ordinal proportional cost sharing,, Journal of Mathematical Economics, 37 (2002), 215. doi: 10.1016/S0304-4068(02)00016-2.

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##### References:
 [1] A. Amirteimoori and S. Kordrostami, Allocating fixed costs and target setting: A DEA-based approach,, Applied Mathematics and Computation, 171 (2005), 136. doi: 10.1016/j.amc.2005.01.064. [2] J. E. Beasley, Allocating fixed costs and resources via data envelopment analysis,, European Journal of Operational Research, 147 (2003), 198. doi: 10.1016/S0377-2217(02)00244-8. [3] A. Cadena, A. Marcucci, J. F. Pérez, H. Durán, H. Mutis, C. Taútiva and F. Palacios, Efficiency analysis in electricity transmission utilities,, Journal of Industrial and management optimization, 5 (2009), 253. doi: 10.3934/jimo.2009.5.253. [4] A. Charnes, W. W. Cooper and E. Rhodes, Measuring the efficiency of decision making units,, European Journal of Operational Research, 2 (1978), 429. doi: 10.1016/0377-2217(78)90138-8. [5] W. D. Cook and M. Kress, Characterizing an equitable allocation of shared costs: A DEA approach,, European Journal of Operational Research, 119 (1999), 652. doi: 10.1016/S0377-2217(98)00337-3. [6] W. D. Cook and J. Zhu, Allocation of shared costs among decision making units: A DEA approach,, Computers & Operations Research, 32 (2005), 2171. doi: 10.1016/j.cor.2004.02.007. [7] W. W. Cooper, L. M. Seiford and K. Tone, Data Envelopment Analysis,, 2nd edition, (2007). [8] 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. [9] G. R. Jahanshahloo, F. Hosseinzadeh Lotfi, N. Shoja and M. Sanei, An alternative approach for equitable allocation of shared costs by using DEA,, Applied Mathematics and Computation, 153 (2004), 267. doi: 10.1016/S0096-3003(03)00631-3. [10] M. Khodabakhshi and K. Aryavash, The fair allocation of common fixed cost or revenue using DEA concept,, Annals of Operational Research, 214 (2014), 187. doi: 10.1007/s10479-012-1117-2. [11] 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. doi: 10.1016/j.ejor.2008.06.017. [12] 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. doi: 10.1016/j.omega.2011.02.008. [13] R. Lin, Allocating fixed costs or resources and setting targets via data envelopment analysis,, Applied Mathematics and Computation, 217 (2011), 6349. doi: 10.1016/j.amc.2011.01.008. [14] R. Lin, Allocating fixed costs and common revenue via data envelopment analysis,, Applied Mathematics and Computation, 218 (2011), 3680. doi: 10.1016/j.amc.2011.09.011. [15] M. Mahdiloo, A. Noorizadeh and R. Farzipoor Saen, Developing a new data envelopment analysis model for custer value analysis,, Journal of Industrial and management optimization, 7 (2011), 531. [16] A. Z. Milioni, J. V. G. Avellar, E. G. Gomes and J. C. B. Soares de Mello, An ellipsoidal frontier model: Allocating input via parametric DEA,, European Journal of Operational Research, 209 (2011), 113. doi: 10.1016/j.ejor.2010.08.008. [17] A. Z. Milioni, E. C. C. Guedes, J. V. G. Avellar and R. C. Silva, Adjusted spherical frontier model: Allocating input via parametric DEA,, Journal of the Operational Research Society, 63 (2012), 406. [18] H. Moulin and R. Stong, Fair queuing and other probabilistic allocation methods,, Mathematics of Operations Research, 27 (2002), 1. doi: 10.1287/moor.27.1.1.336. [19] 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. doi: 10.1016/j.amc.2012.12.085. [20] R. C. Silva and A. Z. Milioni, The adjusted spherical drontier model with weight restrictions,, European Journal of Operational Research, 220 (2012), 729. doi: 10.1016/j.ejor.2012.01.064. [21] T. Sueyoshi and K. Sekitani, An occurrence of multiple projections in DEA-based measurement of technical efficiency: Theoretical comparison among DEA models from desirable properties,, European Journal of Operational Research, 196 (2009), 764. doi: 10.1016/j.ejor.2008.01.045. [22] Y. T. Wang and D. X. Zhu, Ordinal proportional cost sharing,, Journal of Mathematical Economics, 37 (2002), 215. doi: 10.1016/S0304-4068(02)00016-2.
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