Using lateral transshipments can be beneficial in order to improve service levels and to reduce system costs. This paper deals with a single-echelon inventory system with two identical locations. Demands are generated by stationary and independent Poisson processes. In general, if a demand occurs at a location and there is no stock on hand, the demand is assumed to be backordered or lost. However, in this paper, lateral transshipments serve as an emergency supply in case of stock out. In this paper, the rule for lateral transshipments is given, while the ordering policies for normal replenishments are optimized. The transshipment rule is to always transship when there is a shortage at one location and stock on hand at the other. First, we assume that the locations apply (R,Q)(R,Q) policies for normal replenishments, and show that the optimal policies are not necessarily symmetric even though the locations are identical. This means that one cannot in general assume that the optimal policy is symmetric under symmetric assumptions. Second, we relax the assumption of (R,Q)(R,Q) policies and derive the optimal replenishment policy using stochastic dynamic programming.
Normally, in divergent inventory systems, installations at some lower echelon replenish stock from a central warehouse at some upper echelon. By introducing the possibility of transshipment of stock between installations at the lower echelon, better service and reduced system cost can be obtained. Consider for instance the case when an installation has no stock on hand when a demand occurs. Then, instead of replenishing stock from the central warehouse with relatively long leadtime, a transshipment from another installation is realized to meet this demand and thus obtain better service. This sort of transshipments is often termed emergency transshipments, since the call for the transshipment originated from a shortage.
Instead of making transshipments when an installation faces a shortage or has no stock on hand, transshipments can be made routinely as a balancing act in order to reduce the system cost. Transshipment policies of this type have often been applied in periodic review models.
Different from most previous literature, we consider a model in which there is a set-up cost for replenishment. In most previous papers it is assumed that set-up costs are negligible, which implies that order-up-to inventory policies are reasonable, see e.g., Axsäter (1990) or Lee (1987). In this paper we derive the optimal replenishment policy under a given transshipment rule, but also examines how well the commonly used (R,Q)(R,Q) policy performs compared to the optimal policy.
One of the first papers which mentioned the transshipment problem was Clark and Scarf (1960). However, they ignored the problem due to the mathematical complexity. In another early paper, Krishnan and Rao (1965) develop a periodic review, single-echelon model in which they allow transshipments between the lower echelon stock facilities. Gross (1963) considers a transshipment model where it is assumed that both ordering and transshipments are made before demand is realized.
Some papers which present results on optimality are Das (1975), Robinson (1990), Archibald et al. (1997), Herer and Rashit (1999), and Axsäter (2003).
Das (1975) considers a two-location, single-echelon, and single-period problem with periodic review. The main contribution of Das is the opportunity to transfer stock in the middle of a period. He derives optimal transfer and ordering rules by using stochastic dynamic programming, and shows that the so-called complete pooling policy is optimal.
Robinson (1990) studies a multi-period and multi-location problem where transshipments between locations are possible. Under the assumptions of negligible transshipment- and replenishment leadtimes, he demonstrates the optimality of the order-up-to policy. However, analytical results are only found in the two-location case. In the general case, Robinson suggests a heuristic solution method.
Archibald et al. (1997) examine an inventory system much related to Robinson (1990). One difference is that in Archibald et al. transshipments can be made at any time during the period. They formulate the problem as a Markov decision process, and allow emergency orders from the external supplier if a transshipment from another location is not possible. Also in this paper all leadtimes are assumed to be zero. Clearly, it is a major limitation to assume that replenishments occur instantaneously since transshipment and replenishment policies in general depend on both stock on hand and orders in transit.
Herer and Rashit (1999) study transshipment with fixed and joint replenishment costs, but only with single period.
A quite recent paper by Minner and Silver (2005) considers a distribution system with two identical locations, in which lateral transshipments are allowed. The rule for lateral transshipments is, however, not optimized. The locations apply (R,Q)(R,Q) policies, and demand occurs according to a compound Poisson process. They assume that all unsatisfied demand after transshipments is lost, and develop heuristics in order to being able to evaluate costs.
We have presented a model of a single-echelon, two-location system where lateral transshipments are allowed. Most earlier papers which consider lateral transshipments deal with (S-1,S)(S-1,S) policies. In this paper, we generalize the ordering policy and derive the optimal policy by stochastic dynamic programming. We also evaluate the performance of the commonly used (R,Q)(R,Q) policy, in order to see how well the (R,Q)(R,Q) policy approximates the optimal policy. In our numerical study we find that an (R,Q)(R,Q) policy may be a reasonable choice of replenishment policy when the demand rate is relatively low.
Another interesting feature we found is that the optimal policy of the class of (R,Q)(R,Q) policies and the overall optimal policy are not always symmetric even though the locations are identical. This is an interesting structural result, since then we cannot in general assume that the locations should have the same optimal replenishment policy when they are identical.
