The proposed model derives closed-form solutions for investment efforts in both product development and its associated production processes while balancing economic profit, product quality, and marginal production costs. The system dynamics, including state and control variables, as well as the relevant constraints, are explicitly formulated. To demonstrate the practical utility of the framework, a numerical example is investigated. Simulation results illustrate adaptive strategies that anticipate future market conditions, manage product–process innovation trade-offs, and respond effectively to changing innovation returns. In the baseline numerical scenario, the model achieves a cumulative profit of 7850 with an average period profit of 196. Finally, a sensitivity analysis is conducted to examine the impact of key model parameters on system performance.
Mejjaouli et al. (Thu,) studied this question.