Most periodic review models in the inventory literature have assumed a fixed length of the review periods. In this note, we extend the work of Chiang (2008) , and consider backlogged and lost-sales periodic review models where the review periods are of a variable length and there is a fixed cost of ordering for replenishment. Assuming that period lengths are independently and identically distributed, we show (using an exact method of computing inventory holding costs) that an (s, S) policy is optimal for the infinite horizon problem. The periodic review policies developed are thus easy to implement. The computation shows that if the fixed cost of ordering is small, one needs to use the proposed periodic policies.
Most periodic review systems in the inventory-control literature have assumed a fixed length of the review periods. It is possible in practice that periodic systems have the review periods of a variable length. Such systems arise mainly from supply uncertainties. For example, Chiang (2008) observed that many supermarkets have suppliers who come to visit regularly and replenish inventories for them. However, the supplier does not always come in constant time intervals. Depending on her visit plans or work schedules, she often arrives at a particular supermarket earlier or later than planned. The elapsed time between two consecutive visits varies in nature. See also Ertogral and Rahim (2005) for supply chain settings where the replenishment epochs are not under the retailer′s control (i.e., under the supplier′s control), and Tang and Musa (2011) for a variety of supply chain risks or uncertainties.
To the best of our knowledge, the issue of the period length variability or replenishment interval randomness is investigated only recently by Ertogral and Rahim (2005) and Chiang (2008). Ertogral and Rahim (2005) derived the expected profit per replenishment cycle by assuming constant demand; Chiang (2008) used dynamic programming to develop periodic review inventory models with stochastic demand. However, these studies assumed that the fixed cost of ordering for replenishment is zero.
In this paper, we extend the work of Chiang (2008) and incorporate a fixed cost of ordering. It is possible that the supplier visits a retailer and charges a service expense if the retailer′s inventory is replenished. Moreover, instead of using an approximate method as in Ertogral and Rahim (2005) and Chiang (2008), we use an exact approach of computing inventory holding costs. We assume that period lengths are independently and identically distributed (iid), as in the above two studies, and examine both the backlogged and lost-sales periodic review inventory problems. We will show that the optimal policy is of the (s, S) type. Hence, existing algorithms (e.g., Zheng and Federgruen, 1991) could be used to find the optimal s and S. The periodic review models developed can be viewed as a generalization of ordinary periodic models where the period length is fixed.
The computation shows that when the fixed cost of ordering is small (but not small enough to be neglected, such that an order is always placed at a review epoch), ignoring the period length variability can incur unnecessary large losses, especially if lead-time is zero, shortage is costly, demand variability is small, and/or the period length is volatile. These results agree with Chiang (2008). Hence, one needs to use the proposed ordering policies, and the suggestions made in Chiang (2008) apply here, e.g., if the replenishment epochs are under the supplier′s control, the retailer should somehow persuade the supplier to visit more regularly, or even cooperate or form a strategic alliance with the supplier in the long run; Prajogo et al. (2012) recently showed that strategic long-term relationship, one of the three supplier management practices suggested, has a positive relationship with a firm′s operational performance, and Cheng et al. (2012) found that a purchasing firm tends to form quanxi networks with its key supplier to improve communication and thus reduce supply risk.. However, when the fixed ordering cost is large, ignoring the period length variability causes small or virtually no losses to a firm, especially if lead-time is long or demand variability is large. The implication of this is that it is alright to use the ordinary periodic review models in the case of large fixed ordering costs.
In this paper, we consider periodic inventory models where the review periods are of a variable length and there is a fixed cost of ordering for replenishment. Assuming that period lengths are independently and identically distributed, we show that an (s, S) policy is optimal for the infinite horizon problem. Hence, existing algorithms could be used to obtain the optimal s and S. The periodic review inventory policies developed in this paper are thus easy to implement.
The computation shows that when the fixed ordering cost is small, ignoring the period length variability can incur large costs if lead-time is zero, shortage is costly, and/or demand variability is small. It also shows that a firm is more vulnerable to the period length variability if the period length is volatile. These results agree with those in Chiang (2008); hence, the suggestions made in Chiang (2008) apply here.
The computation also shows that when the fixed ordering cost is large, ignoring the period length variability incurs insignificant or no loss, particularly if lead-time is long or demand variability is large. This means that one need not use the proposed periodic review models in the case of large fixed ordering costs.
