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This paper reconsiders the lost-sales inventory system studied by Hill (2007). The commonly assumed policy to apply to the system is a pure base-stock policy (PBSP) for which the best base stock is easily found. Hill shows that his simple delay policy (SDP) and full delay policy (FDP) perform better. The SDP is a (s,d) policy where s is the base stock of the best PBSP and d is a common lower bound on the delay between the placement of successive replenishment orders. We show by simulation that the d value suggested by Johansen (2001) outperforms Hill's suggestion and that the performance often can be further improved by optimizing d. For the test bed investigated by Hill, we show that, for some parameter settings, an additional improvement is achieved when s and d are optimized simultaneously. The policy suggested by Johansen performs better than the FDP in all settings where the former policy reduces the average cost of the best PBSP by at least 1%.

We reconsider the lost-sales inventory system studied by Hill (2007). The system has Poisson demand with rate λλ, continuous review and a fixed lead time L . There is a holding cost h per unit per unit time and a penalty p per unit lost. The objective is to minimize the long-run average cost per unit time subject to the condition that all replenishments are unit sized. This condition is met without loss of optimality when, as assumed by Hill, there is no fixed order cost because then economies of scale are lacking. However, unless λλ is relatively big, a good replenishment policy satisfying the condition for a positive fixed order cost can be found if p is computed as the difference between the lost sales cost per unit and the fixed order cost. The considered system can describe slow-moving but important and possibly expensive spare parts for which the replenishment lead time is relatively long. When demand for such parts occurs during a stockout, the demand is lost to the regular control system because it is satisfied (at an extra cost) by some other means. For the retail sector, the system can describe high-value goods for which a customer demand is lost if the item is not in stock. Both applications often have Poisson demand with a rate λλ which is not relatively big. Then a continuous review model provides a reasonable representation of the system. The commonly assumed policy to apply to the system is a pure base-stock policy (PBSP). It prescribes to maintain the inventory position (the sum of the stock in hand and the stock on order) at some base-stock level s . Hence, if the initial inventory position equals the chosen s , then a new replenishment order for one unit is placed immediately whenever a demand is satisfied. As explained in Section 2 it is straightforward to find the best base stock sPBSPsPBSP for the PBSP. However, Hill (1999) has shown that a PBSP can never be optimal if sPBSP>1sPBSP>1, which applies for most realistic parameter settings. Johansen (2001) and Hill (2007) offer better solutions by suggesting modified base-stock policies (MBSPs) which impose some minimum delay between the placement of successive replenishment orders. The intuitive reason for such a minimum is that, if we shortly after placing an order place another one, then we are very likely to be in stock when the second order arrives and therefore we would be increasing stock with a very small likelihood of that unit being immediately needed. In this paper we investigate by simulation how the considered system performs when it is controlled by different MBSPs specified by a pair (s,d), where s is the base stock and the lower delay bound is fixed as d. The investigated policies are related to (and some of them improve) the MBSPs suggested by Johansen and Hill. The paper is organized as follows. Section 2 provides a brief review of related literature. Our simulation models of the MBSPs are presented in Section 3. Numerical results obtained by the simulation models are reported and discussed in Section 4 and Section 5 contains our conclusion.

We have investigated by simulation how the inventory system studied by Hill (2007) performs when it is controlled by three modified base-stock policies (MBSPs) suggested by us rather than by the simple delay policy (SDP) or the full delay policy (FDP) suggested by Hill. The SDP and our MBSPs were specified by a pair (s,d), where s is a base stock and d is a lower bound on the delay between the placement of successive replenishment orders, whereas the FDP computes a suggested delay each time a sale occurs or an order is placed. For the most interesting part of the test bed used by Hill, our numerical study showed the simulated cost reductions obtained by the investigated policies relative to the cost of the best pure base-stock policy (PBSP). We have observed that the MBSP with s as the best base stock for the PBSP and d as lower delay bound specified by Eq. (2) performs better than the FDP in all parameter settings where the former policy reduces the cost of the PBSP by at least 1%. No simulation is needed to specify this MBSP. It is our favorite because it is easy to specify and implement. However, for some parameter settings, further relative cost reductions can be obtained when d is isolated or the pair (s,d) are simultaneously optimized. But optimization increases the computational burden substantially.