In the design of tolerance allocation the cost-tolerance function is usually employed to represent the objective function which is to be minimized. The traditional cost-tolerance functions in the literature are concerned with only one characteristic. In this paper we obtain a bivariate cost-tolerance function to describe the relationship between the cost and tolerances of two characteristics (i.e. the thickness and inner diameter) of a lock wheel. Then the bivariate loss function is combined with the bivariate cost-tolerance function to determine the optimal tolerances for the thickness and inner diameter of a lock wheel such that the user's potential loss/cost may be evaluated. When the quality loss is considered, the tolerances of both characteristics become tighter. By including the effect of product degradation, the present work of expected bivariate quality loss is then introduced as a quality performance measure. By assuming linear drifts on both the thickness and inner diameter of the lock wheels, the model with the present worth of quality loss leads to tighter tolerances of both characteristics. In addition, a longer planning horizon (or a longer useful life of the product) leads to tighter tolerances and a larger user's discount rate results in looser tolerances for both characteristics.