Optimal patching policy for warrantied smart products considering user adoption and delayed installation

Smart products integrating hardware and software components are increasingly common. Although pre-release testing improves reliability, residual software faults often remain after launch and may cause failures in the post-release phase. To mitigate these risks, developers typically issue patches and provide warranty services. However, most existing studies assume prompt patch adoption and installation, an assumption that rarely holds in practice. This study develops a patching policy for warrantied smart products that explicitly accounts for user adoption and delayed installation behaviour. User response is modelled as a two-stage stochastic process: adoption follows Bernoulli trials with type-specific Beta-distributed probabilities, while installation delays follow user-type-dependent distributions. Users are classified into active and passive groups to capture behavioural heterogeneity. The model incorporates patch coverage effects on fault detection and hardware–software interaction in failure risk. The objective is to determine the optimal number of patches that minimises expected total warranty cost. A Monte Carlo–based algorithm is developed, with analytical results under a simplified setting. Numerical results show that user behaviour significantly affects optimal patching decisions and warranty costs, and that optimal policies may differ markedly from those under full and immediate patch installation. These findings offer managerial insights for designing robust patching policies under behavioural uncertainty.