Optimal classification of failures in a two-unit series system

A two-unit series system with arbitrary distributions for the life-time of the units is considered. Whenever the system fails, it is brought to a repair shop where it is classified as to which unit has failed. In the case of mis-classification, there is cost incurred on erroneous repair of a good unit. Based on the life-time of the system an optimal classification rule is suggested so that the expected loss due to mis-classification is minimized. Two examples are included to illustrate the result.