Maximizing Economic Benefit: A Case Study of Royalty Payment Optimization using Modified Shooting and Discretization Methods

Abstract

The shooting method is reviewed as a numerical solution for addressing non-standard optimal control (OC) problems. The non-standard OC problem is taken into account when the final state value’s component is unknown and free. As a result, the final shadow value, known as the costate variable, was not equal to zero. The objective function also involves the royalty function, which takes the form of a piecewise function. At a certain time frame, it is, nevertheless, not differentiable. So, to determine the unknown final state value, a new modified shooting method was applied. The model could be differentiated at all times thanks to the simultaneous use of a continuous hyperbolic tangent (tanh) approximation. The Sufahani-Ahmad-Powell-Golden-Royalty Algorithm (SAPGRA) was used to construct the problem in C++ programming language. The findings that meet the optimality criteria were then contrasted with discretization methods such as Euler, Runge-Kutta, Trapezoidal and Hermite-Simpson approximation. This groundbreaking discovery is immensely helpful in resolving practical issues. It can advance the academic discipline so that problem-solving techniques are always current. Meanwhile, this study also discusses the value of fundamental theory in solving real-world economic problems.