DISPOSICIONES ADICIONALES

For the single-location inventory system with multiple shipment modes, early papers by Barankin (1961), Daniel (1962), Neuts (1964), Bulinskaya (1964) and Veinott (1966) study optimal ordering policies for periodic review inventory sys- tems with a nominal lead time of one period and an emergency lead time of zero periods. Fukuda (1964) and Wright (1969) allow for multi-period nomi- nal lead times; but, the lead time of the emergency order is always one period shorter than that of nominal order lead time. Rosenshine and Obee (1976) ex- amine the effectiveness of a standing order inventory system (fixed-size order arrives at the beginning of each period), allowing a zero lead time emergency order if large shortage occurs and sell-offs if inventory exceeds storage capac- ity. Whittemore and Saunders (1977) derive the optimal policy for an infinite horizon inventory system allowing two types of orders with multi-period nom- inal lead times. Blumenfeld et al. (1985) analyze the trade-off between safety stock (inventory cost) and emergency order penalties assuming that the emer- gency order is placed only once in a review period, when a shortage occurs, and arrives immediately with a sufficient amount to meet any demands before the next regular shipment.

papers of Gross and Soriano (1972) and Chiang and Gutierrez (1996) analyze the decision rule for choosing the shipment mode for the whole resupply order which needs to be placed. Chiang and Gutierrez (1996) consider emergency re- supply lead times which are less than one review period in length. In a sequel, Chiang and Gutierrez (1998) study a different inventory system with emergency orders placed on a continuous basis while regular orders are placed periodically. Tagaras and Vlachos (2001) propose a simple approximate model for a similar type of inventory system assuming that during each cycle period, the emer- gency orders may be issued only once at specific review epochs. Teunter and Vlachos (2001) allow for emergency orders to arrive some fixed number of time units before a regular resupply order is due. This could be considered as a gen- eralization of the emergency order conditions of the former two papers. Zhang (1996), Feng et al.(2005, 2006) discuss the optimality of the base-stock policy with multiple delivery modes in different situations.

For periodic review multi-stage supply chain systems with emergency or- ders, the paper of Lawson and Porteus (2000) studies a serial supply chain sys- tem allowing expedited shipment with zero lead time between any adjacent echelons, where the regular lead time is always one period. Muharremo ˘glu and Tsitsiklis (2003) extend the model of Lawson and Porteus (2000) by allowing complete lead time flexibility with a more general cost structure for the expe- dited shipping units from any stage in the system to any downstream stage. Huggins and Olsen (2003, 2005) examine optimal policies for two-stage supply chain system under both centralized and decentralized control where expedit- ing can be used to satisfy unmet demand in each period.

D’Esopo (1968) propose three operational parameters: reorder point, order quantity, and expediting level. They propose that whenever the inventory level drops to the expediting level, an outstanding order will be expedited and de- livered after a constant period which is shorter than the constant regular lead time. Muckstadt and Thomas (1980) analyze multi-item multi-echelon inven-

tory systems with (S − 1, S ) inventory policy and Poisson demand considering

emergency orders as lost sales. Moinzadeh and Nahmias (1988) propose an ex- tension of the (Q, R) policy with different reorder points and reorder sizes for regular and emergency orders. In deriving cost expressions, they assume that there is never more than one outstanding order of each type. Assuming Poisson demand, Moinzadeh and Schmidt (1991) consider a single location (S-1,S) in- ventory system with two options (regular and emergency) resupply. Emergency orders arrive in a shorter time but at a higher cost. Assuming that lead times are known and constant, they propose an order policy incorporating the age of the outstanding orders. Moinzadeh and Aggarwal (1997) adopt the same class of policies but extend the results to a multi-echelon inventory system. Considering the standard (Q, R) policy for regular replenishment orders with constant lead time, Johansen and Thorstenson (1998) let the emergency orders (with short, constant lead times) depend also on the remaining delivery time for a regular order. Chiang (2002) proposes two single-location, single-item policies when expediting is allowed. One policy modifies the Allen and D’Esopo (1968) pol- icy by adding a threshold time point, which is the last point when expediting is allowed, while the other policy allows an expediting decision to be made at some point during the lead time. The problem of using multiple suppliers effi- ciently is reviewed by Minner (2003). Axs¨ater (2005) considers a single-echelon continuous review inventory system facing compound Poisson demand. The

system gets normal replenishments according to a standard (Q, R) policy and emergency replenishments in critical situations. Axs¨ater suggests a heuristic decision rule for triggering emergency orders, which minimizes the expected costs under the assumption that there is only at most one emergency replenish- ment outstanding at any time.

An extensive literature is also available studying lateral transshipments among locations. Many papers examine the effect of employing decision rules for making lateral transshipments. Examples of this type are Lee (1987), Axs¨ater (1990), Dada (1992), Alfredsson and Verrijdt (1999), Grahovac and Chakravarty (2001), Kukreja et al. (2001), Sherbrooke (1992), Muckstadt (2005), Vidgren (2005), Axs¨ater (2006) and Vliegen (2009). Other papers present methods for optimizing the decisions concerning lateral transshipments. Examples are Das (1975), Robinson (1990), Tagaras and Cohen (1992), Archibald et al. (1997), Rudi

et al. (2001), Minner et al. (2003), Wong et al. (2006), Olsson (2009), Kranenburg

and van Houtum (2009), Wijk et al. (2009) and Reijnen et al.(2009). Paterson

et al. (2009) provide an up-to-date review of the inventory models with lateral

transshipment.

The system considered in this paper differs from others in the literature in that it explicitly considers a stocking location that is dedicated to satisfying emergency orders, and attempts to estimate the distribution of outstanding or- ders for this type of location. For the purposes of analysis, we assume con-

tinuous review (S − 1, S ) stocking policy Stuttering Poisson demand and expo-

nentially distributed lead times. We also investigate constant lead times in an empirical study.

3.2

Emergency Orders with Compound Poisson Demand and