Parallel time scales and two-dimensional manufacturer and individual customer warranties

We deal with a system whose failures depend on several parallel effects, such as the time in use ℒ and the mileage ℋ. Manufacturer warranties are typically described by a two-dimensional region in the (ℒ, ℋ)-plane. A proper determination of the warranty limits must be based on a two-dimensional distribution of time to failure on this plane. The aim of this paper is to demonstrate the possibility of designing individual warranties for a “nontypical” customer who has a very low or very high usage rate b = H/L, and to show a simple way to calculate warranty limits by minimizing the lifetime coefficient of variation. The latter is carried out by introducing the “best” combined time scale in the form  = (1−ε)ℒ+εℋ which provides the minimal lifetime coefficient of variation.