This paper presents a method for determining the optimal two-dimensional warranty region for age and mileage scales across all possible combinations of free replacement warranty (FRW), prorata warranty (PRW), and FRW-PRW combined policies. The operational time or lifetime and usage of the products are modeled using a bivariate Gumbel copula with Weibull as the marginal distribution. The optimal warranty region is derived by maximizing an expected utility function, which incorporates two cost components: the economic benefit function and the warranty cost function, specifically constructed for the two-dimensional warranty scenario. To obtain the optimal warranty region, a real-world dataset of traction motors, including age and mileage information, is analyzed. The results show that considering a two-dimensional warranty cost function yields the highest utility compared to all other scenarios.
Das et al. (2025) studied this question.