Multi-Objective Optimization of a Multi-Server Retrial Machine Repair System with Orbital Search and Synchronous Vacation

This paper investigates a multi-server retrial machine repair system that incorporates orbital search and a synchronous vacation mechanism. The system features standby units and examines two potential vacation scenarios for servers, reflecting real-world situations, such as technical staff in teaching hospitals taking periodic administrative or training vacations. We formulate a mathematical model using birth-and-death processes to establish the governing equation and propose a recursive matrix method to systematically derive the steady-state probabilities. Performance measures, including system availability, the expected number of failed units in the orbit, and expected waiting times, are derived. To address the conflicting objectives of minimizing total operating costs, minimizing expected waiting times, and maximizing system availability, we construct a tri-objective optimization problem. By implementing a multi-objective genetic algorithm, we identify a set of Pareto-optimal frontiers and reveal the explicit financial and operational trade-offs among these competing criteria. Numerical experiments and sensitivity analyses demonstrate that enhancing the automated retrial rate and managing the emergency repair rate are most critical to minimizing system downtime. Furthermore, the joint optimization of server capacities and vacation schedules effectively eliminates operational redundancy, showing that near-perfect equipment availability (up to 0.999) can be achieved with only marginal increases in cost. This research provides administrators with a robust decision-making framework to optimize technical resource management while ensuring near-perfect equipment availability in real-world environments.