Bozulan sistemlerde bileşenlerin optimal değişim zamanı

Optimal replacement time problems involve determining the best time to undertake a replacement action within a stochastic system to either minimize associated cost or maximize related rewards of the process, based on limited information. This problem can be encountered in many different application areas such as healthcare, machine maintenance, finance, manufacturing etc. Identifying the appropriate intervention time is crucial for preventing system degradation that may ultimately lead to failure. This study investigates an optimal stopping problem focused on determining the optimal point to replace a component which belongs to a stochastically degrading system. The system's functionality is maintained by replacing deteriorated components with heterogeneous spares of varying quality. The decision framework includes two actions: replace and wait. The objective is to identify the optimal switching point between these decisions to maximize the system's expected reward. The problem is formulated as a Markov Decision Process (MDP) with an average reward criterion. The structural properties of the optimal value function are analyzed to demonstrate the existence of an optimal threshold-type policy. Subsequently, this policy is derived by exploiting the Markovian dynamics of the system. Two alternative methods are proposed to calculate the optimal threshold point in addition to the Value Iteration algorithm and they are implemented in Python to compute optimal solutions across different input parameter settings, illustrating the model's adaptability and robustness. Numerical results are analyzed to assess the sensitivity of system parameters. Additionally, extensions related to system states and quality of spare parts are also included to the model to see their effect on system behavior. The findings offer a comprehensive framework for addressing optimal replacement problems.