A note on 'A bivariate optimal replacement policy for a repairable system'

An error appearing in equation (3) of Y.L. Zhang ( J. Appl. Prob., 1994, 31, 1123-1127) has been pointed out by S.H. Sheu ( Eur. J. Oper. Res., 1999, 112, 503-516) and the correct expressions (25)-(27) given accordingly on pp. 510-511. However, the derivation of the key expression (27), the long-run expected loss rate, was not presented. The purpose of this note is threefold. First, since a monotone process (e.g. an arithmetic, geometric, or arithmetic-geometric process) approach, as discussed by K.N.F. Leung ( Eng. Optimiz., 2001, 33, 473-484), is considered to be relevant, realistic, and appropriate to the modelling of a deteriorating system maintenance problem, it is worth explicitly developing this expression, which is of benefit to the subsequent studies. Secondly, equation (3) in Zhang (1994) is shown to be fundamentally correct and so it can be viewed as an alternative method of formulating similar bivariate cases. Thirdly, although equations (4) and (5) in Zhang (1994) have been logically and correctly derived, both can be readily reduced to their simplest forms which are derived here.