مدل تولید، توزیع یکپارچه برای یک آیتم موجودی رو به وخامت

We develop an integrated production–distribution model for a deteriorating item in a two-echelon supply chain. The supplier’s production batch size is restricted to an integer multiple of the discrete delivery lot quantity to the buyer. Exact cost functions for the supplier, the buyer and the entire supply chain are developed. These lead to the determination of individual optimal policies, as well as the optimal policy for the overall, integrated supply chain. We outline a procedure for determining the optimal supply chain decisions with the objective of minimizing the total system cost. Our approach is illustrated through a numerical example.

In real life, it is not uncommon for inventory items, such as milk, fruit, blood, pharmaceutical product, vegetables etc, to decay or deteriorate over time. Therefore, it is important to study the behavior of such decaying and deteriorating items, towards the formulation of appropriate inventory control policies that explicitly take such behavior into account. Ghare and Schrader (1963) were the first authors to consider the effect of decay on inventory items. They used the term “inventory decay” to describe this phenomenon, including direct spoilage, physical depletion and deterioration. They developed a general EOQ type model under constant demand with exponential decay, which could be solved iteratively, but not directly. Covert and Philip (1973) and Tadikamalla (1978) extended Ghare and Schrader’s work to Weibull and gamma distribution based deterioration patterns, respectively. All of the above models assumed instantaneous replenishment and did not allow backorders or shortages. Misra (1975) developed EOQ type models with finite production rates, but did not allow backorders. His models include cases of varying and constant deterioration rates. Shah (1977) developed models for both exponential and Weibull deterioration cases, which allowed backorders albeit with instantaneous replenishment. Mak (1982) developed a production lot size inventory model with backorders for exponentially decaying items. Heng et al. (1991) integrated Misra’s and Shah’s approaches and developed a model for deteriorating items with finite production rate and backorders. Deterioration inventory model research has received increasing attentions in recent years, extending existing models with a variety of deterioration patterns, demand functions and backordering policies. In most cases, deterioration is assumed to be a constant fraction of total on-hand inventory. The Weibull distribution has been used to model item decay (Chakrabarty et al., 1998), and some attention has been focused on deteriorating items with expiration dates (Hsu et al., 2006 and Lo et al., 2007). Recent work in this area has considered time-varying as well as stock level and price dependent demands. Wee, 1993 and Wee, 1995, Hariga et al. (1997), Bhunia and Maiti (1998), Chung and Tsai (2001), Wang (2002), Balkhi (2004), Teng and Chang (2005), Yang (2005), Shah et al. (2005), Chang et al. (2006), Hou, 2006 and Hou, 2006, Wu et al. (2006), Manna and Chaudhuri (2006), Pal et al. (2006) and Lo et al. (2007) provide good examples of inventory models with different assumptions concerning the patterns of demand, deterioration and backordering. Most existing inventory models for deteriorating items are EOQ type models and consider the different sub-systems in the supply chain independently. Although the notion of cooperation between suppliers and buyers has received more and more attention in the literature, few integrated inventory approaches for deteriorating items have been developed to date. Rau et al. (2003) develop a multi-echelon inventory model for a deteriorating item and derive the optimal joint total cost from an integrated perspective, including the supplier, the producer and the buyer. Yang and Wee (2003) develop a mathematical model for multi-item production lot sizing for deteriorating items. As mentioned above, the existing literature suggests a variety of inventory models pertaining to deteriorating items. Few of them, however, have been developed explicitly for the JIT environment. Yang and Wee (2003) developed an integrated multi-product lot-size inventory model for deteriorating items in JIT environment. Both the lot-splitting of material from the supplier to producer, and the lot-splitting of finished good from producer to multiple buyers are considered. However, the optimal solutions are not given. Furthermore, the differential equations used in their paper are not in point any more when the number of buyers is small, such as 1. This paper considers the single-supplier, single-buyer case, involving a deteriorating inventory item in a JIT environment. Almost all authors to date use calculus based approaches for solving the inventory models developed for deteriorating items. Such approaches become more complicated in the single-supplier, single-buyer case when cost functions for the supplier and buyer are derived separately. In fact, when a lot is delivered from the supplier to the buyer, at specified “delivery points”, the inventory level of the supplier changes suddenly and forms an inflexion on the supplier’s inventory function. These inflexions make it difficult to use classical optimization techniques. With the common assumption that the item’s deterioration rate is small and its square or higher powers can be neglected, this paper intends to derive the cost functions for the suppler, buyer and the entire supply chain and derive appropriate policies based on an algebraic method. In short, this paper intends to fill a notable gap in the existing literature concerning inventory models for deteriorating items in JIT environments. It may also be seen as an extension and generalization of the recent work by Kim and Ha (2003).

In this study, we have developed an integrated single-supplier, single-buyer inventory model for a deteriorating item in a JIT environment. Cost functions for the supplier, the buyer and the integrated supply chain are derived. This paper also discusses the optimal interval for the number of deliveries per production batch cycle, which is assumed to be an integer. Based on the notion of this optimal interval, we outline an effective procedure for determining the optimum delivery lot size to the buyer, the supplier’s optimal production batch size and number of deliveries resulting from each such batch. Our results show that when the deterioration rate increases, both the optimal production lot size and the cycle time decrease (see Table 1 and Fig. 3). This is not unexpected, because when the deterioration rate goes up, the production lot size and the corresponding cycle time are reduced for the benefit of the entire supply chain. It is interesting to note that an increase in the deterioration rate also tends to reduce the delivery lot size without affecting the number of deliveries per production batch. In fact, as seen in Table 1, the optimal interval for N, which does not appear to be unmanageably large, does not change with varying deterioration rates beyond d=0. The reductions in the inventory cycle times for both the parties, thus, appear to mitigate the negative effects of deterioration on the supply chain. Not surprisingly, our results also indicate that, the average annual cost for the supply chain increases as the deterioration rate goes up (see Table 1 and Fig. 4). This occurs due to the impact of deterioration on the inventory cycles, i.e. more frequent setups and deliveries than would otherwise be necessary without deterioration. Finally, it would be interesting for future work to study the effects of item decay allowing backorders in an integrated supply chain context.