Optimal age for preventive replacement under a combined fully renewable free replacement with a pro-rata warranty

This paper focuses on an age-replacement policy for products under a new warranty strategy, which combines a fully renewable free replacement with a pro-rata warranty policy (called fully renewable FRW/PRW policy). For this combined warranty strategy, whenever a product fails within the warranty period, it is replaced by a new one and a new warranty is issued. Meanwhile, if the failure occurs during the first time interval (i.e., the free-replacement warranty (FRW) interval), replacement is made at no cost to the buyer; however, if the failure occurs during the second time interval (i.e., the pro-rata warranty (PRW) interval), replacement is made at a pro-rata cost to the buyer. Dividing the replacement age into three separate periods: within FRW, during PRW, and post-warranty. Then, given that the replacement age is in a certain interval, the cost model from the user/buyer perspective is developed, and the corresponding local optimal replacement age is derived such that the long run expected cost rate is minimized. Afterward, the global optimal replacement age is determined. Structural properties of these optimal policies are obtained and presented, and special cases of the model are discussed. Finally, a numerical example is given for illustration.