New fundamental theory in solving the royalty payment problem / Wan Noor Afifah Wan Ahmad and Suliadi Firdaus Sufahani

This study deal with the non-classical Optimal Control problem (OCP) where we want to maximize the functional objective function. However, this project has to clarify a few constraints. Firstly, the final state value is unknown. Furthermore, the objective function involved the royalty function which is in terms of the piecewise function of the unknown final state value. In addition, the non-classical OCP of the unknown final state value resulting in the nonzero shadow value at the final time. Moreover, the royalty function is non-differentiable at a certain process. Therefore, this study applied a new modified shooting method which is Sufahani-Ahmad-Newton-Golden-Royalty Algorithm. The program is constructed in the C++ program language. As a validation process, the result was compared with the discretization method which is constructed in AMPL program language with MINOS solver. The discretization method involved Euler, Runge-Kutta, Trapezoidal, and Hermite- Simpson methods. It is expected that the new modified shooting method will give a more accurate optimal solution when compared with the discretization method. The difficulty in using the royalty function that is in the piecewise function is overcome by utilizing the continuous approximation of the hyperbolic tangent (tanh) approach. This is to make sure that the objective function can be differentiated at all processes. This project can be a stepping-stone for the researcher to explore a new approach in solving a real-world problem. The most important is, this project addressed the importance of fundamental theory in solving the untangled issue.