Optimal replacement strategies for warranty products with multiple failure modes after warranty expiry

The optimal replacement problem for a warranty product subject to (M + 1) types of mutually exclusive failure modes, including M repairable failure modes and a catastrophic failure mode, is investigated in this paper. When the product fails with a repairable failure mode, a minimal repair will be conducted to remove the failure. However, if a catastrophic failure mode occurs, the failure product can only be replaced by a new one. Within the warranty period, the seller needs to complete each failure repair within a repair time limit. Once a failure cannot be repaired within the stipulated time, under the rebate warranty, the customer is refunded a proportion of the sale price of the product based on its working time, and the warranty terminates simultaneously; or under the free replacement warranty, the seller is obligated to provide a new product at no charge to the customer. Under both the warranty policies, the repair cost is free of charge to the customer within the warranty time, while the customer is responsible for all the cost for repairs within the post-warranty time. From the customer's point of view, the expected life cycle cost rate of the product under the rebate and free replacement warranty are respectively derived. The existence and uniqueness of the optimal replacement strategies after warranty expiring are also proved theoretically by minimizing the cost rate. Finally, we give numerical examples to illustrate the two warranty models and show the effectiveness of proposed strategies.