This research delves into the application of the modified shooting method for the numerical resolution of non-standard optimal control (OC) problems. More precisely, it concentrates on scenarios where the final state value component remains unknown and unconstrained, leading to a non-zero final shadow value or costate variable. Moreover, the objective function involved a piecewise royalty function, which poses a challenge due to its lack of differentiability within a specific time interval. Consequently, the novel modified shooting method was employed to ascertain the elusive final state value. The model’s differentiability is maintained throughout by adopting a continuous hyperbolic tangent (tanh) approximation. In addition, the construction of the Sufahani-Ahmad-Newton-Golden-Royalty Algorithm (SANGRA) and Sufahani-Ahmad-Powell-Golden-Royalty Algorithm (SAPGRA) was accomplished using the C++ programming language to formulate the problem. The outcomes of these algorithms, satisfying the criteria for optimality, were juxtaposed with non-linear programming (NLP) techniques such as Euler and Runge-Kutta, aside from Trapezoidal and Hermite-Simpson approximations. This groundbreaking discovery carries extensive practical implications as it propels the field forward and ensures the application of contemporary problem-solving methodologies. Moreover, the study underscores the significance of fundamental theory in effectively tackling current economic challenges.

discretization method, non-standard optimal control, optimality condition, royalty payment problem, modified shooting method.

Received: August 13, 2024; Accepted: September 18, 2024; Published: October 16, 2024

How to cite this article: Wan Noor Afifah Wan Ahmad, Suliadi Firdaus Sufahani, Mahmod Abd Hakim Mohamad and Norhaslinda Zainal Abidin, Optimizing royalty payments for maximum economic benefit: a case study utilizing modified shooting and discretization methods, Advances in Differential Equations and Control Processes 31(4) (2024), 563-581. https://doi.org/10.17654/0974324324029

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References:

[1] Y. J. Cai, Y. Chen, T. Siqin, T. M. Choi and S. H. Chung, Pay upfront or pay later? Fixed royal payment in sustainable fashion brand franchising, International Journal of Production Economics 214 (2019), 95-105.
https://doi.org/10.1016/j.ijpe.2019.03.025
[2] P. A. F. Cruz, D. F. M. Torres and A. S. I. Zinober, A non-classical class of variational problems, International Journal of Mathematical Modelling and Numerical Optimization 1(3) (2010), 227-236.
https://doi.org/10.1504/IJMMNO.2010.031750
[3] R. Fourer, D. M. Gay and B. W. Kernighan, A modelling language for mathematical programming, Management Science 36(5) (1990), 519-554.
https://doi.org/10.1287/mnsc.36.5.519
[4] D. M. Gay, The AMPL modeling language: An aid to formulating and solving optimization problems, Proceedings of the Numerical Analysis and Optimization: NAO-III, Muscat, Oman, January 2014, Springer International Publishing (2015), 95-116. https://doi.org/10.1007/978-3-319-17689-5_5
[5] L. T. Horal, I. V. Perevozova and V. I. Shyiko, Actualization definitions and theoretical justification distribution ratio of oil and gas royalties under decentralization, Scientific Bulletin of Ivano-Frankivsk National Technical University of Oil and Gas (Series: Economics and Management in the Oil and Gas Industry) 2(20) (2019), 21-32.
[6] D. E. Kirk, Optimal Control Theory: An Introduction, Courier Corporation, 2004.
[7] A. B. Malinowska and D. F. M. Torres, Natural boundary conditions in the Calculus of Variations, Mathematical Methods in the Applied Sciences 33(14) (2010), 1712-1722. https://doi.org/10.1002/mma.1289
[8] B. Passenberg, M. Kröninger, G. Schnattinger, M. Leibold, O. Stursberg and M. Buss, Initialization concepts for optimal control of hybrid systems, Proceedings of the 18th World Congress the International Federation of Automatic Control, Elsevier 44(1) (2011), 10274-10280.
https://doi.org/10.3182/20110828-6-IT-1002.03012
[9] A. M. Spence, The learning curve and competition, The Bell Journal of Economics 12(1) (1981), 49-70. https://doi.org/10.2307/3003508
[10] O. V. Stryk and R. Bulirsch, Direct and indirect methods for trajectory optimization, Annals of Operational Research 37(1) (1992), 357-373.
https://doi.org/10.1007/BF02071065
[11] A. Yahya and A. Habbal, Music royalty payment scheme using blockchain technology, Proceedings of the 5th International Symposium on Multidisciplinary Studies and Innovative Technologies, IEEE (2021), 539-545.
https://doi.org/10.1109/ISMSIT52890.2021.9604559
[12] A. Zinober and K. Kaivanto, Optimal Production Subject to Piecewise Continuous Royalty Payment Obligations, University of Sheffield, 2008.
[13] A. S. I. Zinober and S. F. Sufahani, A non-standard optimal control problem arising in an economics application, Pesquisa Operacional 33(1) (2013), 63-71. https://doi.org/10.1590/S0101-74382013000100004.