Fixed Cost Allocation Based on Explainable AI with Cross-Efficiency Approach: DEA–Game Enhanced by Decision Tree

Authors

  • Masoumeh Raeiszadeh * Department of Industrial Engineering, Najafabad Branch, Islamic Azad University, Najafabad, Iran.
  • Javad Gerami Department of Mathematics, Shi.C., Islamic Azad University, Shiraz, Iran. https://orcid.org/0000-0001-9013-1634

https://doi.org/10.48314/ijorai.v2i1.82

Abstract

The issue of fixed cost allocation in organizations with heterogeneous Decision-Making Units (DMUs) has always been accompanied by challenges related to fairness, managerial acceptance, and the stability of results. This research presents an innovative hybrid framework for fixed cost allocation by extending and generalizing the Data Envelopment Analysis (DEA)–Game Cross Efficiency approach introduced by Li et al. [1], explicitly considering the structural heterogeneity of DMUs. The primary innovation of this study is the integration of a Decision Tree algorithm as a preprocessing step, which categorizes DMUs into homogeneous subgroups based on input–output patterns, thereby enhancing the validity of efficiency comparisons within the DEA framework. After segregating the units, within each homogeneous subgroup, cross-efficiency DEA is first calculated, followed by leveraging the DEA–Game Cross Efficiency model to define the characteristic function of the cooperative game based on cross-efficiency improvements. Subsequently, using the Shapley value as the unique solution to the cooperative game, the fair share of fixed costs for each unit is determined. The proposed framework theoretically possesses the property of superadditivity, making the formation of a full coalition of units rational and stable. The efficiency and implementability of the proposed method were tested through a real-world numerical example involving the allocation of 10 trillion Rials in advertising costs among 8 DMUs in a dairy company. The numerical results demonstrated that the proposed method leads to a significantly different and more meaningful distribution compared to traditional methods and even the baseline model of Li et al. [1], such that the cost share of each unit directly aligns with its marginal role in increasing collective efficiency. These findings indicate that the proposed framework is not only mathematically fairer but also has higher managerial acceptability and applicability in real-world heterogeneous environments.   

Keywords:

Fixed cost allocation, Data envelopment analysis–game cross efficiency, Decision Tree, Shapley value

References

  1. [1] Li, F., Zhu, Q., & Liang, L. (2018). Allocating a fixed cost based on a DEA-game cross efficiency approach. Expert systems with applications, 96, 196–207. https://doi.org/10.1016/j.eswa.2017.12.002
  2. [2] Charnes, A., Cooper, W. W., & Rhodes, E. (1978). Measuring the efficiency of decision making units. European journal of operational research, 2(6), 429–444. https://doi.org/10.1016/0377-2217(78)90138-8
  3. [3] Cook, W. D., & Seiford, L. M. (2009). Data envelopment analysis (DEA)-thirty years on. European journal of operational research, 192(1), 1–17. https://doi.org/10.1016/j.ejor.2008.01.032
  4. [4] Emrouznejad, A., Parker, B. R., & Tavares, G. (2008). Evaluation of research in efficiency and productivity: A survey and analysis of the first 30 years of scholarly literature in DEA. Socio-economic planning sciences, 42(3), 151–157. https://doi.org/10.1016/j.seps.2007.07.002
  5. [5] Cook, W. D., & Kress, M. (1999). Characterizing an equitable allocation of shared costs: A DEA approach. European journal of operational research, 119(3), 652–661. https://doi.org/10.1016/S0377-2217(98)00337-3
  6. [6] Beasley, J. E. (2003). Allocating fixed costs and resources via data envelopment analysis. European journal of operational research, 147(1), 198–216. https://doi.org/10.1016/S0377-2217(02)00244-8
  7. [7] Amirteimoori, A., & Mohaghegh Tabar, M. (2010). Resource allocation and target setting in data envelopment analysis. Expert systems with applications, 37(4), 3036–3039. https://doi.org/10.1016/j.eswa.2009.09.029
  8. [8] Cook, W. D., & Zhu, J. (2005). Allocation of shared costs among decision making units: A DEA approach. Computers & operations research, 32(8), 2171–2178. https://doi.org/10.1016/j.cor.2004.02.007
  9. [9] Lin, R., & Peng, Y. (2011). A fixed cost allocation approach with dea super efficiency invariance. 2011 international conference on electronics, communications and control (ICECC) (pp. 622–625). IEEE. https://doi.org/10.1109/ICECC.2011.6066601
  10. [10] Pishvaee, M. S., Rabbani, M., & Torabi, S. A. (2011). A robust optimization approach to closed-loop supply chain network design under uncertainty. Applied mathematical modelling, 35(2), 637–649. https://doi.org/10.1016/j.apm.2010.07.013
  11. [11] Doyle, J., & Green, R. (1994). Efficiency and cross-efficiency in DEA: Derivations, meanings and uses. Journal of the operational research society, 45(5), 567–578. https://doi.org/10.1057/jors.1994.84
  12. [12] Adler, N., Friedman, L., & Sinuany-Stern, Z. (2002). Review of ranking methods in the data envelopment analysis context. European journal of operational research, 140(2), 249–265. https://doi.org/10.1016/S0377-2217(02)00068-1
  13. [13] Xu, G., Wu, J., Zhu, Q., & Pan, Y. (2024). Fixed cost allocation based on data envelopment analysis from inequality aversion perspectives. European journal of operational research, 313(1), 281–295. https://doi.org/10.1016/j.ejor.2023.08.020
  14. [14] Liang, L., Wu, J., Cook, W. D., & Zhu, J. (2008). The DEA game cross-efficiency model and its Nash equilibrium. Operations research, 56(5), 1278–1288. https://doi.org/10.1287/opre.1070.0487
  15. [15] Wu, J., Liang, L., & Yang, F. (2009). Determination of the weights for the ultimate cross efficiency using Shapley value in cooperative game. Expert systems with applications, 36(1), 872–876. https://doi.org/10.1016/j.eswa.2007.10.006
  16. [16] An, Q., Wang, P., Emrouznejad, A., & Hu, J. (2020). Fixed cost allocation based on the principle of efficiency invariance in two-stage systems. European journal of operational research, 283(2), 662–675. https://doi.org/10.1016/j.ejor.2019.11.031
  17. [17] Breiman, L. (1996). Bagging predictors. Machine learning, 24(2), 123–140. https://doi.org/10.1007/BF00058655
  18. [18] Quinlan, J. R. (1986). Induction of decision trees. Machine learning, 1(1), 81–106. https://doi.org/10.1007/BF00116251
  19. [19] Safavian, S. R., & Landgrebe, D. (1991). A survey of decision tree classifier methodology. IEEE transactions on systems, man, and cybernetics, 21(3), 660–674. https://doi.org/10.1109/21.97458

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Published

2026-03-05

Issue

Vol. 2 No. 1 (2026): International Journal of Operations Research and Artificial Intelligence (IJORAI)

Section

Articles

How to Cite

Raeiszadeh, M. ., & Gerami, J. . (2026). Fixed Cost Allocation Based on Explainable AI with Cross-Efficiency Approach: DEA–Game Enhanced by Decision Tree. International Journal of Operations Research and Artificial Intelligence , 2(1), 1-10. https://doi.org/10.48314/ijorai.v2i1.82

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