# American Institute of Mathematical Sciences

October  2020, 7(4): 303-315. doi: 10.3934/jdg.2020024

## On the grey Baker-Thompson rule

 1 Süleyman Demirel University, Faculty of Engineering, Department of Industrial Engineering, Isparta, 32260, Turkey 2 Süleyman Demirel University, Faculty of Economics and Administrative Sciences, Department of Business Administration, Isparta, 32260, Turkey 3 Süleyman Demirel University, Faculty of Arts and Sciences, Department of Mathematics, Isparta, 32260, Turkey

* Corresponding author: zeynepalparslan@yahoo.com

Received  December 2018 Revised  April 2020 Published  October 2020 Early access  August 2020

Cost sharing problems can arise from situations in which some service is provided to a variety of different customers who differ in the amount or type of service they need. One can think of and airports computers, telephones. This paper studies an airport problem which is concerned with the cost sharing of an airstrip between airplanes assuming that one airstrip is sufficient to serve all airplanes. Each airplane needs an airstrip whose length can be different across airplanes. Also, it is important how should the cost of each airstrip be shared among airplanes. The purpose of the present paper is to give an axiomatic characterization of the Baker-Thompson rule by using grey calculus. Further, it is shown that each of our main axioms (population fairness, smallest-cost consistency and balanced population impact) together with various combina tions of our minor axioms characterizes the best-known rule for the problem, namely the Baker-Thompson rule. Finally, it is demonstrated that the grey Shapley value of airport game and the grey Baker-Thompson rule coincides.

Citation: Mehmet Onur Olgun, Osman Palanci, Sirma Zeynep Alparslan Gök. On the grey Baker-Thompson rule. Journal of Dynamics and Games, 2020, 7 (4) : 303-315. doi: 10.3934/jdg.2020024
##### References:
 [1] S. Z. A. Gök, On the interval Baker-Thompson rule, J. Appl. Math., 2012, Article ID 218792, 5 pp. doi: 10.1155/2012/218792. [2] S. Z. A. Gök, R. Branzei and S. Tijs, Convex interval games, J. Appl. Math. Dec. Sci., 2009, Article ID 342089, 14 pp. doi: 10.1155/2009/342089. [3] S. Z. A. Gök, R. Branzei and S. Tijs, Airport interval games and their Shapley value, Oper. Res. Dec., 19 (2009b), 9-18. [4] A. Çevik, G.-W. Weber and B. M. Eyüboğlu, Voxel-MARS: A method for early detection of Alzheimer's disease by classification of structural brain MRI, Ann. Oper. Res., 258 (2017), 31-57.  doi: 10.1007/s10479-017-2405-7. [5] Y. Chun, C. C. Hu and C. H. Yeh, Characterizations of the sequential equal contributions rule for the airport problem, Internat. J. Econom. Theory, 8 (2012), 77-85.  doi: 10.1111/j.1742-7363.2011.00175.x. [6] J. Baker Jr., Airport Runway Cost Impact Study, Report submitted to the Association of Local Transport Airlines, Jackson, Mississippi, 1965. [7] I. D. Baltas and A. N. Yannacopoulos, Uncertainty and inside information, J. Dynam. Games, 3 (2016), 1-24.  doi: 10.3934/jdg.2016001. [8] S. K. Das, S. K. Roy and G.-W. Weber, An exact and a heuristic approach for the transportation-p-facility location problem, Comput. Manag. Sci., (2020). doi: 10.1007/s10100-019-00610-7. [9] J. Deng, Control problems of Grey systems, Syst. Contr. Lett., 5 (1982), 288-294.  doi: 10.1016/S0167-6911(82)80025-X. [10] J. Deng, Grey System Fundamental Method, Huazhong University of Science and Technology, China, 1985. [11] V. Fragnelli and M. E. Marina, An axiomatic characterization of the Baker-Thompson rule, Econom. Lett., 107 (2010), 85-87.  doi: 10.1016/j.econlet.2009.12.033. [12] M. Ghoreishi, G.-W. Weber and A. Mirzazadeh, An inventory model for non-instantaneous deteriorating items with partial backlogging, permissible delay in payments, inflation- and selling price-dependent demand and customer returns, Ann. Oper. Res., 226 (2015), 221-238.  doi: 10.1007/s10479-014-1739-7. [13] S. Khalilpourazari, A. Mirzazadeh, G.-W. Weber and S. H. R. Pasandideh, A robust fuzzy approach for constrained multi-product economic production quantity with imperfect items and rework process, Optimization, 69 (2020), 6-90.  doi: 10.1080/02331934.2019.1630625. [14] E. Kropat, G.-W. Weber and S. Belen, Dynamical gene-environment networks under ellipsoidal uncertainty - Set-theoretic regression analysis based on ellipsoidal OR, in Dynamics, Games and Science I, Springer Proceedings in Mathematics, 1, Springer Berlin-Heidelberg, 2011,545–571. doi: 10.1007/978-3-642-11456-4_35. [15] S. C. Littlechild and G. Owen, A simple expression for the Shapley value in a special case, Manag. Sci., 20 (1973), 370-372.  doi: 10.1007/BF01766216. [16] S. C. Littlechild and G. F. Thompson, Aircraft landing fees: A game theory approach, Bell J. Econom., 8 (1977), 186-204. [17] S. Liu and Y. Lin, Grey Information: Theory and Practical Applications, Springer, Germany, 2006. [18] R. Lotfi, G.-W. Weber, S. M. Sajadifar and N. Mardani, Interdependent demand in the two-period newsvendor problem, J. Indust. Manag. Optim., 16 (2020), 117-140.  doi: 10.3934/jimo.2018143. [19] R. B. Myerson, Conference structures and fair allocation rules, Internat. J. Game Theory, 9 (1980), 169-182.  doi: 10.1007/BF01781371. [20] A. Özmen, G.-W. Weber, I. Batmaz and E. Kropat, RCMARS: Robustification of CMARS with different scenarios under polyhedral uncertainty set, Commun. Nonlin. Sci. Numer. Simul., 16 (2011), 4780-4787.  doi: 10.1016/j.cnsns.2011.04.001. [21] O. Palancı, S. Z. A. Gök, S. Ergün and G.-W. Weber, Cooperative grey games and grey Shapley value, Optimization, 64 (2015), 1657-1668.  doi: 10.1080/02331934.2014.956743. [22] T. Paksoy, E. Özceylan and G.-W. Weber, Profit oriented supply chain network optimization, Cent. Eur. J. Oper. Res., 21 (2013), 455-478.  doi: 10.1007/s10100-012-0240-0. [23] E. Savku and G.-W. Weber, A stochastic maximum principle for a Markov regime-switching jump-diffusion model with delay and an application to finance, J. Optim. Theory Appl. Springer, 179 (2018), 696-721.  doi: 10.1007/s10957-017-1159-3. [24] E. Qasım, S. Z. A. Gök, O. Palancı and G.-W. Weber, Airport situations and games with grey uncertainty, Internat. J. Indust. Eng. Oper. Res., 1 (2019), 51-59. [25] B. Z. Temoçin and G.-W. Weber, Optimal control of stochastic hybrid system with jumps: A numerical approximation, J. Comput. Appl. Math., 259 (2014), 443-451.  doi: 10.1016/j.cam.2013.10.021. [26] S. Tijs, Introduction to Game Theory, Hindustan Book Agency, India, 2003. [27] E. B. Tirkolaee, A. Goli and G.-W. Weber, Multi-objective aggregate production planning model considering overtime and outsourcing options under fuzzy seasonal demand, in Advances in Manufacturing II. MANUFACTURING 2019. Lecture Notes in Mechanical Engineering, Springer, Cham, 2019. [28] G. F. Thompson, Airport Costs and Pricing, Unpublished Ph.D Dissertation, University of Birmingham, 1971. [29] G.-W. Weber, Ö. Uğur, P. Taylan and A. Tezel, On optimization, dynamics and uncertainty: A tutorial for gene-environment networks, Netw. Comput. Biol. Disc. Appl. Math., 157 (2009), 2494-2513.  doi: 10.1016/j.dam.2008.06.030. [30] F. Yerlikaya-Özkurt and P. Taylan, New computational methods for classification problems in the existence of outliers based on conic quadratic optimization, Commun. Statist.-Sim. Comput., 49 (2020), 753-770.  doi: 10.1080/03610918.2019.1661477.

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##### References:
 [1] S. Z. A. Gök, On the interval Baker-Thompson rule, J. Appl. Math., 2012, Article ID 218792, 5 pp. doi: 10.1155/2012/218792. [2] S. Z. A. Gök, R. Branzei and S. Tijs, Convex interval games, J. Appl. Math. Dec. Sci., 2009, Article ID 342089, 14 pp. doi: 10.1155/2009/342089. [3] S. Z. A. Gök, R. Branzei and S. Tijs, Airport interval games and their Shapley value, Oper. Res. Dec., 19 (2009b), 9-18. [4] A. Çevik, G.-W. Weber and B. M. Eyüboğlu, Voxel-MARS: A method for early detection of Alzheimer's disease by classification of structural brain MRI, Ann. Oper. Res., 258 (2017), 31-57.  doi: 10.1007/s10479-017-2405-7. [5] Y. Chun, C. C. Hu and C. H. Yeh, Characterizations of the sequential equal contributions rule for the airport problem, Internat. J. Econom. Theory, 8 (2012), 77-85.  doi: 10.1111/j.1742-7363.2011.00175.x. [6] J. Baker Jr., Airport Runway Cost Impact Study, Report submitted to the Association of Local Transport Airlines, Jackson, Mississippi, 1965. [7] I. D. Baltas and A. N. Yannacopoulos, Uncertainty and inside information, J. Dynam. Games, 3 (2016), 1-24.  doi: 10.3934/jdg.2016001. [8] S. K. Das, S. K. Roy and G.-W. Weber, An exact and a heuristic approach for the transportation-p-facility location problem, Comput. Manag. Sci., (2020). doi: 10.1007/s10100-019-00610-7. [9] J. Deng, Control problems of Grey systems, Syst. Contr. Lett., 5 (1982), 288-294.  doi: 10.1016/S0167-6911(82)80025-X. [10] J. Deng, Grey System Fundamental Method, Huazhong University of Science and Technology, China, 1985. [11] V. Fragnelli and M. E. Marina, An axiomatic characterization of the Baker-Thompson rule, Econom. Lett., 107 (2010), 85-87.  doi: 10.1016/j.econlet.2009.12.033. [12] M. Ghoreishi, G.-W. Weber and A. Mirzazadeh, An inventory model for non-instantaneous deteriorating items with partial backlogging, permissible delay in payments, inflation- and selling price-dependent demand and customer returns, Ann. Oper. Res., 226 (2015), 221-238.  doi: 10.1007/s10479-014-1739-7. [13] S. Khalilpourazari, A. Mirzazadeh, G.-W. Weber and S. H. R. Pasandideh, A robust fuzzy approach for constrained multi-product economic production quantity with imperfect items and rework process, Optimization, 69 (2020), 6-90.  doi: 10.1080/02331934.2019.1630625. [14] E. Kropat, G.-W. Weber and S. Belen, Dynamical gene-environment networks under ellipsoidal uncertainty - Set-theoretic regression analysis based on ellipsoidal OR, in Dynamics, Games and Science I, Springer Proceedings in Mathematics, 1, Springer Berlin-Heidelberg, 2011,545–571. doi: 10.1007/978-3-642-11456-4_35. [15] S. C. Littlechild and G. Owen, A simple expression for the Shapley value in a special case, Manag. Sci., 20 (1973), 370-372.  doi: 10.1007/BF01766216. [16] S. C. Littlechild and G. F. Thompson, Aircraft landing fees: A game theory approach, Bell J. Econom., 8 (1977), 186-204. [17] S. Liu and Y. Lin, Grey Information: Theory and Practical Applications, Springer, Germany, 2006. [18] R. Lotfi, G.-W. Weber, S. M. Sajadifar and N. Mardani, Interdependent demand in the two-period newsvendor problem, J. Indust. Manag. Optim., 16 (2020), 117-140.  doi: 10.3934/jimo.2018143. [19] R. B. Myerson, Conference structures and fair allocation rules, Internat. J. Game Theory, 9 (1980), 169-182.  doi: 10.1007/BF01781371. [20] A. Özmen, G.-W. Weber, I. Batmaz and E. Kropat, RCMARS: Robustification of CMARS with different scenarios under polyhedral uncertainty set, Commun. Nonlin. Sci. Numer. Simul., 16 (2011), 4780-4787.  doi: 10.1016/j.cnsns.2011.04.001. [21] O. Palancı, S. Z. A. Gök, S. Ergün and G.-W. Weber, Cooperative grey games and grey Shapley value, Optimization, 64 (2015), 1657-1668.  doi: 10.1080/02331934.2014.956743. [22] T. Paksoy, E. Özceylan and G.-W. Weber, Profit oriented supply chain network optimization, Cent. Eur. J. Oper. Res., 21 (2013), 455-478.  doi: 10.1007/s10100-012-0240-0. [23] E. Savku and G.-W. Weber, A stochastic maximum principle for a Markov regime-switching jump-diffusion model with delay and an application to finance, J. Optim. Theory Appl. Springer, 179 (2018), 696-721.  doi: 10.1007/s10957-017-1159-3. [24] E. Qasım, S. Z. A. Gök, O. Palancı and G.-W. Weber, Airport situations and games with grey uncertainty, Internat. J. Indust. Eng. Oper. Res., 1 (2019), 51-59. [25] B. Z. Temoçin and G.-W. Weber, Optimal control of stochastic hybrid system with jumps: A numerical approximation, J. Comput. Appl. Math., 259 (2014), 443-451.  doi: 10.1016/j.cam.2013.10.021. [26] S. Tijs, Introduction to Game Theory, Hindustan Book Agency, India, 2003. [27] E. B. Tirkolaee, A. Goli and G.-W. Weber, Multi-objective aggregate production planning model considering overtime and outsourcing options under fuzzy seasonal demand, in Advances in Manufacturing II. MANUFACTURING 2019. Lecture Notes in Mechanical Engineering, Springer, Cham, 2019. [28] G. F. Thompson, Airport Costs and Pricing, Unpublished Ph.D Dissertation, University of Birmingham, 1971. [29] G.-W. Weber, Ö. Uğur, P. Taylan and A. Tezel, On optimization, dynamics and uncertainty: A tutorial for gene-environment networks, Netw. Comput. Biol. Disc. Appl. Math., 157 (2009), 2494-2513.  doi: 10.1016/j.dam.2008.06.030. [30] F. Yerlikaya-Özkurt and P. Taylan, New computational methods for classification problems in the existence of outliers based on conic quadratic optimization, Commun. Statist.-Sim. Comput., 49 (2020), 753-770.  doi: 10.1080/03610918.2019.1661477.
The flowchart of this study
Grey marginal vectors
 $\sigma$ $m_{1}^{\sigma }\left(c^{\prime }\right)$ $m_{2}^{\sigma }\left(c^{\prime }\right)$ $m_{3}^{\sigma }\left(c^{\prime }\right)$ $\sigma _{1} = \left(1, 2, 3\right)$ $m_{1}^{\sigma _{1}}\left(c^{\prime }\right) \in \left[30, 36\right]$ $m_{2}^{\sigma _{1}}\left(c^{\prime }\right) \in \left[14, 18\right]$ $m_{3}^{\sigma _{1}}\left(c^{\prime }\right) \in \left[16, 26\right]$ $\sigma _{2} = \left(1, 3, 2\right)$ $m_{1}^{\sigma _{2}}\left(c^{\prime }\right) \in \left[30, 36\right]$ $m_{2}^{\sigma _{2}}\left(c^{\prime }\right) \in \left[0, 0\right]$ $m_{3}^{\sigma _{2}}\left(c^{\prime }\right) \in \left[30, 44\right]$ $\sigma _{3} = \left(2, 1, 3\right)$ $m_{1}^{\sigma _{3}}\left(c^{\prime }\right) \in \left[0, 0\right]$ $m_{2}^{\sigma _{3}}\left(c^{\prime }\right) \in \left[44, 54\right]$ $m_{3}^{\sigma _{3}}\left(c^{\prime }\right) \in \left[16, 26\right]$ $\sigma _{4} = \left(2, 3, 1\right)$ $m_{1}^{\sigma _{4}}\left(c^{\prime }\right) \in \left[0, 0\right]$ $m_{2}^{\sigma _{4}}\left(c^{\prime }\right) \in \left[44, 54\right]$ $m_{3}^{\sigma _{4}}\left(c^{\prime }\right) \in \left[16, 26\right]$ $\sigma _{5} = \left(3, 1, 2\right)$ $m_{1}^{\sigma _{5}}\left(c^{\prime }\right) \in \left[0, 0\right]$ $m_{2}^{\sigma _{5}}\left(c^{\prime }\right) \in \left[0, 0\right]$ $m_{3}^{\sigma _{5}}\left(c^{\prime }\right) \in \left[60, 80\right]$ $\sigma _{6} = \left(3, 2, 1\right)$ $m_{1}^{\sigma _{6}}\left(c^{\prime }\right) \in \left[0, 0\right]$ $m_{2}^{\sigma _{6}}\left(c^{\prime }\right) \in \left[0, 0\right]$ $m_{3}^{\sigma _{6}}\left(c^{\prime }\right) \in \left[60, 80\right]$
 $\sigma$ $m_{1}^{\sigma }\left(c^{\prime }\right)$ $m_{2}^{\sigma }\left(c^{\prime }\right)$ $m_{3}^{\sigma }\left(c^{\prime }\right)$ $\sigma _{1} = \left(1, 2, 3\right)$ $m_{1}^{\sigma _{1}}\left(c^{\prime }\right) \in \left[30, 36\right]$ $m_{2}^{\sigma _{1}}\left(c^{\prime }\right) \in \left[14, 18\right]$ $m_{3}^{\sigma _{1}}\left(c^{\prime }\right) \in \left[16, 26\right]$ $\sigma _{2} = \left(1, 3, 2\right)$ $m_{1}^{\sigma _{2}}\left(c^{\prime }\right) \in \left[30, 36\right]$ $m_{2}^{\sigma _{2}}\left(c^{\prime }\right) \in \left[0, 0\right]$ $m_{3}^{\sigma _{2}}\left(c^{\prime }\right) \in \left[30, 44\right]$ $\sigma _{3} = \left(2, 1, 3\right)$ $m_{1}^{\sigma _{3}}\left(c^{\prime }\right) \in \left[0, 0\right]$ $m_{2}^{\sigma _{3}}\left(c^{\prime }\right) \in \left[44, 54\right]$ $m_{3}^{\sigma _{3}}\left(c^{\prime }\right) \in \left[16, 26\right]$ $\sigma _{4} = \left(2, 3, 1\right)$ $m_{1}^{\sigma _{4}}\left(c^{\prime }\right) \in \left[0, 0\right]$ $m_{2}^{\sigma _{4}}\left(c^{\prime }\right) \in \left[44, 54\right]$ $m_{3}^{\sigma _{4}}\left(c^{\prime }\right) \in \left[16, 26\right]$ $\sigma _{5} = \left(3, 1, 2\right)$ $m_{1}^{\sigma _{5}}\left(c^{\prime }\right) \in \left[0, 0\right]$ $m_{2}^{\sigma _{5}}\left(c^{\prime }\right) \in \left[0, 0\right]$ $m_{3}^{\sigma _{5}}\left(c^{\prime }\right) \in \left[60, 80\right]$ $\sigma _{6} = \left(3, 2, 1\right)$ $m_{1}^{\sigma _{6}}\left(c^{\prime }\right) \in \left[0, 0\right]$ $m_{2}^{\sigma _{6}}\left(c^{\prime }\right) \in \left[0, 0\right]$ $m_{3}^{\sigma _{6}}\left(c^{\prime }\right) \in \left[60, 80\right]$
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