Modified Shooting with Discretization Validation for Non-Standard Optimal Control Problem: Case Study for Four-Stage Royalty Payment Problem

Authors

  • Wan Noor Afifah Wan Ahmad Department of Mathematics and Statistics, Faculty of Applied Sciences and Technology, Universiti Tun Hussein Onn Malaysia, Pagoh Educational Hub, 84600 Pagoh, Johor, Malaysia
  • Suliadi Firdaus Sufahani Department of Mathematics and Statistics, Faculty of Applied Sciences and Technology, Universiti Tun Hussein Onn Malaysia, Pagoh Educational Hub, 84600 Pagoh, Johor, Malaysia
  • Mahmod Abd Hakim Mohamad Department of Mechanical Engineering, Center of Diploma, Universiti Tun Hussein Onn Malaysia, Pagoh Educational Hub, 84600 Pagoh, Johor, Malaysia
  • Mohd Saifullah Rusiman Department of Mathematics and Statistics, Faculty of Applied Sciences and Technology, Universiti Tun Hussein Onn Malaysia, Pagoh Educational Hub, 84600 Pagoh, Johor, Malaysia
  • Mohd Zulariffin Md Maarof Department of Sciences and Mathematics, Center of Diploma, Universiti Tun Hussein Onn Malaysia, 84600 Pagoh Educational Hub, Pagoh, Johor, Malaysia
  • Muhamad Ali Imran Kamarudin School of Business Management, College of Business, Universiti Utara Malaysia, 06010 Sintok, Kedah, Malaysia
  • Muhaimin Ismoen School of Applied Sciences and Mathematics, Universiti Teknologi Brunei, Jalan Tungku Link Gadong, BE1410 Brunei Darussalam
  • Lee Kah Howe Ecube Global Pte Ltd, 1 CLaymore Drive, 08-03, Orchard Tower Rear Block APT, 229594 Singapore

DOI:

https://doi.org/10.37934/araset.59.2.296311

Keywords:

Discretization, hyperbolic tangent function, optimal control, royalty problem, shooting method

Abstract

A modified shooting method augmented by discretization validation is presented in this paper to address the challenges inherent in non-standard optimal control problems. Specifically, a specific case study involving four-stage royalty payment functions is focused on, with the aim of effectively optimizing these complex, non-differentiable functions. The shooting method is adapted and enhanced to compute optimal solutions, and its accuracy is rigorously confirmed through discretization techniques. The primary objectives involve maximizing the performance index and determining optimal control strategies for scenarios where the final state variable remains unknown. In this study, the royalty payment is defined as a four-stage piecewise function. The incorporation of piecewise royalty functions introduces non-differentiability at specific time intervals, necessitating innovative approaches for finding optimal solutions. This, in turn, leads to the utilization of the continuous hyperbolic tangent (tanh) function to address the non-differentiability issue. A hybrid shooting method, combining the Newton and Golden Section Search methods, is employed in the C++ programming language to compute the unknown final state value. A new natural boundary condition, based on established theory, is introduced to further facilitate the investigation. Discretization methods such as Euler, Runge-Kutta, Trapezoidal, and Hermite-Simpson approximations are employed for validation. The validation process entails the use of the AMPL programming language with the MINOS solver. Comparative analyses reveal that the modified shooting method yields more accurate optimal results than discretization methods, thus demonstrating the method’s effectiveness in addressing non-standard optimal control problems. The significance of fundamental theories in addressing real-world challenges is underscored by this research, providing valuable insights for future researchers exploring mathematical approaches in similar contexts. The study contributes to the continued relevance of the academic field, particularly in science and mathematics education.

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Author Biography

Suliadi Firdaus Sufahani, Department of Mathematics and Statistics, Faculty of Applied Sciences and Technology, Universiti Tun Hussein Onn Malaysia, Pagoh Educational Hub, 84600 Pagoh, Johor, Malaysia

suliadi@uthm.edu.my

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Published

2024-11-04

How to Cite

Wan Ahmad, W. N. A. ., Sufahani, S. F., Mohamad, M. A. H., Rusiman, M. S., Md Maarof, M. Z. ., Kamarudin, M. A. I. ., Ismoen, M. ., & Lee, K. H. (2024). Modified Shooting with Discretization Validation for Non-Standard Optimal Control Problem: Case Study for Four-Stage Royalty Payment Problem. Journal of Advanced Research in Applied Sciences and Engineering Technology, 296–311. https://doi.org/10.37934/araset.59.2.296311

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