We consider a perishable inventory system in which units in stock have a common deterministic lifetime and units of the same storage-age will fail together if they are not taken by demands. Perishable inventory models with stochastic demands are difficult to analyze (Nahmias, 1982). Early works in this area focused on the issuing policy in blood bank management (Prastacos, 1984) and on the inventory replenishment policy under periodic reviews (Nahmias, 1982). This work is concerned with the replenishment policy for continuous review systems.

The general m-period (i.e., lifetime equals m review periods) periodic review model with zero lead time was investigated by Fries (1975) and Nahmias (1975a). They identified some properties of the optimal ordering policy, for example, the fact that the optimal order quantity decreases by less than one unit when the available inventory of any age increases by one unit (Nabmias, 1978). However, due to the complicated nature of the problem, the exact form of the optimal replenishment policy has not been found for the general rn-period periodic review model. Instead, later efforts have been largely focused on finding and computing approximations of the true optimal policy (Chazan and Gal, 1977; Cohen, 1976; Nahmias, 1975b, 1976, 1978; Nansakumar and Morton, 1993).

The continuous review approach has received less attention in the perishable inventory literature. Weiss (1980) studied a continuous review perishable inventory model with a Poisson demand process and zero lead time. He showed that in the lost sales case, when a lost sales penalty is charged for a demand that cannot be satisfied from the shelf but can be satisfied immediately by placing an order, the (0,S) policy is optimal. If this penalty cost is not charged, the optimal policy is (-1,8). He then derived the cost function for the lost sales model with the (0,S) policy and showed that the cost function is unimodal in S. In the backorder case, Weiss (1980) proved that when the shortage cost is increasing convex in response time, the (s, S) policy is optimal. He pointed out further that the (s, S) policy is still optimal when the demand process is compound Poisson. Liu and Lian (1999) studied a continuous review perishable inventory model with a general renewal demand process. Using the Markov renewal theory, they constructed a closed-form cost function for the (s, S) continuous review model allowing backorders. They then demonstrated analytically that this cost function is monotone or unimodal in s and S, respectively. As a result, the numerical optimization can be easily carried out. Schmidt and Nahmias (1985) considered an (S - 1, S) continuous review perishable inventory model with Poisson demands, a fixed lifetime, and a fixed lead time. They investigated the properties of the base stock level S as a function of different system parameters. The authors also concluded that it is unlikely that optimal ordering policies can be found for general perishable inventory models with positive lead times. Ravichandran (1995) analyzed a non-standard perishable model with a positive random lead time and a Poisson demand process. By assuming that the aging of new stock only begins after the complete depletion of the existing stocks, the author obtained some analytical results on the system performance. Chiu (1995) is the only work available in the literature that treated the positive lead time case for a general perishable model under continuous review. He developed an approximation for the expected outdating of the current order Q. Based on heuristic arguments he then formulated the expected cost function and obtained a (Q, r) ...