Equal Surplus Sharing in Grey Inventory Games

Abstract

This study introduces a model where inventory costs are represented as grey numbers, rather than traditional crisp or stochastic values. Utilizing grey calculus, we reinterpret game-theoretic solutions to address interval uncertainty within cooperative grey inventory games. Building on the works of van den Brink and Funaki (2009) and Olgun et al. (2017). We establish grey equal distribution rules for fair cost allocation. We determine problem parameters to construct a grey inventory game, applying it to three shotgun companies in Turkey. The calculated grey inventory costs and different game-theoretic solutions are presented. This study extends solutions like the Banzhaf value, CIS-value, ENSC- value, and ED- solution by incorporating interval uncertainty. Future research may explore extensions such as grey purchasing costs, stock out allowances, defective goods, and quantity discounts, enhancing the application of grey calculus in cooperative game theory and inventory management.

Keywords

• Grey Numbers
• Cooperative Game Theory
• Inventory Management
• Equal Surplus Sharing Rules

References

1. Alparskan Gök, S. Z., Branzei, R., & Tijs, S. (2011). Big Boss Interval Games. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 19(1), 135–149. https://doi.org/10.1142/s0218488511006927
2. Anily, S., & Haviv, M. (2007). The Cost Allocation Problem for the First Order Interaction Joint Replenishment Model. Operations Research, 55(2), 292–302. https://doi.org/10.1287/opre.1060.0346
3. Brink, R. van den, & Funaki, Y. (2008). Axiomatizations of a Class of Equal Surplus Sharing Solutions for TU-Games. Theory and Decision, 67(3), 303–340. https://doi.org/10.1007/s11238-007-9083-x
4. De, S. K., & Mahata, G. C. (2020). Solution of an imperfect-quality EOQ model with backorder under fuzzy lock leadership game approach. International Journal of Intelligent Systems, 36(1), 421–446. https://doi.org/10.1002/int.22305
5. Driessen, T. S. H., & Funaki, Y. (1991). Coincidence of and collinearity between game theoretic solutions. Operations-Research-Spektrum, 13(1), 15–30. https://doi.org/10.1007/bf01719767
6. Driessen, T. S. H., & Tijs, S. H. (1985). The \ensuremath{\tau }-value, The core and semiconvex games. International Journal of Game Theory, 14(4), 229–247. https://doi.org/10.1007/bf01769310
7. Driessen, T. (1988). Cooperative Games and Examples. In Cooperative Games, Solutions and Applications (pp. 1–12). Springer Netherlands. https://doi.org/10.1007/978-94-015-7787-8\_1
8. Driessen, T., & Funaki, Y. (1994). Reduced game properties of egalitarian division rules for cooperative games. In Operations Research '93 (pp. 126–129). Physica-Verlag HD. https://doi.org/10.1007/978-3-642-46955-8\_33

9. Dror, M., & Hartman, B. C. (2011). Survey of cooperative inventory games and extensions. Journal of the Operational Research Society, 62(4), 565–580. https://doi.org/10.1057/jors.2010.65
10. Funaki, Y. (1998). Dual axiomatizations of solutions of cooperative games. Unpublished Results.
11. Guardiola, L. A., Meca, A., & Puerto, J. (2021). Unitary Owen Points in Cooperative Lot-Sizing Models with Backlogging. Mathematics, 9(8), 869. https://doi.org/10.3390/math9080869
12. Harris, F. W. (1913). How many parts to make at once. Factory, The Magazine of Management, 10(2), 135–136.
13. Kahraman, Ö. U., & Aydemir, E. (2020). A bi-objective inventory routing problem with interval grey demand data. Grey Systems: Theory and Application, 10(2), 193–214. https://doi.org/10.1108/gs-12-2019-0065
14. Karsten, F., Slikker, M., & Borm, P. (2017). Cost allocation rules for elastic single-attribute situations: Cost Allocation Rules for Elastic Single-Attribute Situations. Naval Research Logistics (NRL), 64(4), 271–286. https://doi.org/10.1002/nav.21749
15. Kose, E., Temiz, I., & Erol, S. (2011). Grey System Approach for Economic Order Quantity Models under Uncertainty. Journal of Grey System, 23(1), 71–82.
16. Legros, P. (1986). Allocating joint costs by means of the nucleolus. International Journal of Game Theory, 15(2), 109–119. https://doi.org/10.1007/bf01770979
17. Leng, M., & Parlar, M. (2009). Allocation of Cost Savings in a Three-Level Supply Chain with Demand Information Sharing: A Cooperative-Game Approach. Operations Research, 57(1), 200–213. https://doi.org/10.1287/opre.1080.0528
18. Liu, P., Hendalianpour, A., & Hamzehlou, M. (2021). Pricing model of two-echelon supply chain for substitutable products based on double-interval grey-numbers. Journal of Intelligent & Fuzzy Systems, 40(5), 8939–8961. https://doi.org/10.3233/jifs-201206
19. Liu, S., & Forrest, J. Y.-L. (2010). Grey Systems: Theory and Applications. Springer Verlag.
20. Meca, A. (2004). Inventory games. European Journal of Operational Research, 156(1), 127–139. https://doi.org/10.1016/s0377-2217(02)00913-x
21. Meca, A. (2006). A core-allocation family for generalized holding cost games. Mathematical Methods of Operations Research, 65(3), 499–517. https://doi.org/10.1007/s00186-006-0131-z
22. Meca, A., Fiestras-Janeiro, M. G., Mosquera, M. A., & García-Jurado, I. (2010). Cost sharing in distribution problems for franchise operations. Proceedings of the Behavioral and Quantitative Game Theory: Conference on Future Directions, 1–3. https://doi.org/10.1145/1807406.1807482
23. Mosquera, M. A., García-Jurado, I., & Fiestras-Janeiro, M. G. (2007). A note on coalitional manipulation and centralized inventory management. Annals of Operations Research, 158(1), 183–188. https://doi.org/10.1007/s10479-007-0240-y
24. Moulin, H. (1985). The separability axiom and equal-sharing methods. Journal of Economic Theory, 36(1), 120–148. https://doi.org/10.1016/0022-0531(85)90082-1
25. Olgun, M. O. (2017). İşbirlikçi gri stok oyunları. Süleyman Demirel University.
26. Olgun, M. O., & Aydemir, E. (2021). A new cooperative depot sharing approach for inventory routing problem. Annals of Operations Research, 307(1–2), 417–441. https://doi.org/10.1007/s10479-021-04122-z
27. Olgun, M. O., Özdemir, G., & Alparslan Gök, S. Z. (2017). Gri Stok Modelinin İşbirlikçi Oyun Teorisi İle Maliyet Dağıtımlarının İncelenmesi. Uludağ University Journal of the Faculty of Engineering, 23–34. https://doi.org/10.17482/uumfd.335422
28. Yang, Y., Hu, G., & Spanos, C. J. (2021). Optimal Sharing and Fair Cost Allocation of Community Energy Storage. IEEE Transactions on Smart Grid, 12(5), 4185–4194. https://doi.org/10.1109/tsg.2021.3083882