Analysis of a periodic review inventory control system with perishables having random lifetime
Résumé
We consider an order up to level (T, S) policy for perishable items with random lifetime. We investigate two cases: the first one is the case where excess demand is completely lost and the second one deals with full backorders. Demands arrive according to a Poisson process. The lifetime of each item is assumed to be exponentially distributed and the procurement lead time is constant. We also assume that there is at most one outstanding order at any time. We model this inventory system as a Markov process in which stationary regime can be characterised. This model allows us to get insights on the impact of the model parameters on the overall system performance in terms of costs. Keywords: perishable items; periodic review inventory control; Markov process 1. Introduction Perishable items are one of the most important sources of revenue in grocery industry and food stuff (Deniz 2007). They are also an important source of waste due to their limited lifetime. Lystad, Ferguson, and Alexopoulos (2006) report that about $30 billion are lost due to perishable products in US grocery industry. Consequently, the effect of perishability cannot be disregarded. A large part of investigations analysing perishable inventory systems in the literature assume that the inventory is reviewed continuously and products have stochastic lifetime. However, periodic inventory control policies are used in many practical situations, especially in the case of multi-item, multi-echelon inventory systems, where items are often ordered within a common frequency. For instance, van Donselaar et al. (2010) report that stores that order through an automated store ordering system receive at most one shipment per day from the central warehouse which regroups all the items to be delivered that day. The literature concerning perishable inventory systems can be classified depending on whether the inventory is controlled continuously or periodically and whether products' lifetime is assumed to be constant or stochastic. In almost all models, the FIFO issuing policy is used. Other types of issuing policies exist such as the LIFO issuing policy used by Cohen and Prastacos (1978), and the disposal decision model studied by Martin (1986). Since the present paper does not consider the case of a continuous review inventory system, we refer the reader to Kouki (2010), Karaesmen, Scheller-Wolf, and Deniz (2011) and Nahmias (2011) that deal with such specific inventory policies. For periodic review perishable inventory systems with deterministic lifetime, the problem of finding optimal ordering policy is well known as the fixed-life perishability problem (FLPP) (Nandakumar and Morton 1993). If products cannot be held in stock more than one period, the FLPP is reduced to the known Newsboy problem (Khouja 1999). Van Zyl (1964) proposes a first dynamic programme approach for products having a lifetime of two periods and derives the optimal policy when shortage costs are charged to unsatisfied demand. Nahmias and Pierskalla (1973) build a model where they charge a cost associated with the outdating (perished items) and show that the order quantity for a perishable item is always smaller than the one of a non-perishable item. This work is extended by Fries (1975) and Nahmias (1975) to the case where products have three or more units of lifetime. Nahmias (1975) assumes that the cost of outdating is charged to the period in which the order arrives while Fries (1975) assumes that this cost is charged at the period where outdating occurs. These two models are shown to be identical by Nahmias (1977) when the remaining number of periods in the horizon exceeds the product lifetime. Then, Nahmias (1978) considers a fixed ordering cost and emphasises the difficulty to compute the optimal policy by a multidimensional dynamic programming approach. This arises since the dynamic programming approach needs to track the different ages' categories of items in stock. However, direct computation of an optimal policy turns out to be impractical because of the dimension of the dynamic programme generated by the different age categories. Therefore, several other researches have focused on developing heuristic approximations. Nahmias (1976) considers only two ages: the total old quantity of on-hand inventory