مدل و الگوریتم هیوریستیک در مسئله دوباره پر کردن مشترک با سفارش دهی مجدد و تقاضای مرتبط

In practice, demands for items are often correlated with each other because of cross-selling effects. Traditional inventory models hold the assumption that demands are independent, which may lead to the risk of misestimating the inventory cost of associated items. This paper addresses a multi-item inventory system where demands for some minor items are correlated with that for a major item. Because of the cross-selling, demand for minor items may either be raised by the sales of the major item or be pulled down by the unsatisfied demands for the major item. Thus, a joint inventory policy considering correlations across items should be pursued. Considering general integer policy, we propose a joint replenishment problem (JRP) model with complete backordering and correlated demand. Through the use of derivatives the model is transformed to minimize a cost function with respect to multiples of the major item's order cycle. Given that the optimal solution should offer a trade-off between the ordering cost and the inventory holding cost of each item, we develop a heuristic algorithm by adjusting the replenishment frequencies of minor items to solve the model. The heuristic is tested by simulated numerical examples and shows satisfactory results.

Stochastic demand is an uncertain factor that has been considered in many risk-related models of inventory management. For instance, Luciano et al. (2003) employed value-at-risk (VaR) as a risk measure for inventory management in a static multiperiod framework. Tapiero (2005) studied a specific problem based on risk exposure in inventory management considering shortage cost. Jammernegg and Kischka (2009) proposed a newsvendor model for risk-averse and risk-taking inventory managers pursuing a robust inventory decision. On the other hand, it has been generally recognized that cross-selling effects will lead to demands for items being correlated with each other, which can be depicted by the probability with which an item can be demanded together with an associated item. In fact, cross-selling implies the very frequent phenomenon in retail shops or supermarkets whereby some items are always purchased jointly with a given probability by customers due to their unknown interior associations (Anand et al., 1997 and Kleinberg et al., 1998). Because of cross-selling effects, demand for an item can either be raised by the gained sales of its associated item or be pulled down by shortages in the associated item (Brijs et al., 1999 and Wong et al., 2005). The cross-selling effect can be regarded as another uncertain factor that can also lead to the risk in inventory management when demand cannot be satisfied directly from shelf stock. In a single-item inventory system, when stockouts occur, only an independent item is considered. In this case, a frequent context is that unmet demand during the stockout period can be totally backordered and satisfied in the next replenishment. If all the unmet demands can be backordered, the manager will be confronted with the economic order quantity (EOQ) problem with complete backorders, which has been extensively studied (Grubbström and Erdem, 1999, Cárdenas-Barrón, 2001, Ronald et al., 2004, Sphicas, 2006 and Minner, 2007). Recently, Cárdenas-Barrón (2009) proposed a single-stage economic production quantity (EPQ) model with planned backorders considering the presence of defective products. The author assumes that all backorders are completely satisfied to determine the optimal production batch size and backordered quantity for minimizing the total cost per unit time. Cárdenas-Barrón (2010) further presented a method for determining the optimal lot size and backorders utilizing the two well-known inequalities: the arithmetic–geometric mean inequality (AGM) and the Cauchy–Bunyakovsky–Schwarz (CBS) inequality. The advantage of their method is to optimize the inventory model without derivatives. The topic of the single-item inventory model with backorders has attracted considerable attention in inventory management. However, it is more familiar for multiple items to occur in real-world inventory systems, which makes it inevitable for inventory managers to be confronted with multi-item problems in inventory management. Moreover, demands for items are frequently correlated with each other. Thus, we should take into account multiple items in inventory management by introducing correlated demand into inventory models. The basic formulation of the multi-item inventory model is the so-called joint replenishment problem (JRP) (Goyal, 1973 and Silver et al., 1998). In the JRP, it is assumed that many items are procured from the same producer. Any replenishment can give rise to a major ordering cost, and at the same time a minor ordering cost is always associated with the item replenished. For instance, if we order different types of fish, the transportation will incur a major ordering cost because transportation costs always occur at every replenishment point regardless of what kind of fish we ordered. However, the keeping costs of different types of fish during transportation are only correlated to the specific type of fish and are charged only when the corresponding type of fish is replenished. Hence, in the JRP, many items can share a common major ordering cost if replenished simultaneously, and an item also incurs a minor ordering cost if ordered. Obviously, coordination of replenishments of all items reduces the number of major and minor ordering costs, which will lead to a lower total inventory cost. Furthermore, because supply cycles of all items must be coordinated for convenient communication and scheduling, it is a conventional assumption in the JRP that the time interval between replenishments of an item is a positive integer multiple of a basic order cycle. The traditional JRP has been studied extensively including as heuristic algorithms and global optimal algorithms (Goyal, 1973, Goyal, 1974, Goyal, 1985, Goyal, 1988, Silver, 1976, Goyal and Belton, 1979, Kaspi and Rosenblatt, 1983, Kaspi and Rosenblatt, 1991, Goyal and Deshmukh, 1993, Viswanathan, 1996, Wildeman et al., 1997, Lee and Yao, 2003, Nilsson et al., 2007 and Porras and Dekker, 2008), where independent demand and no stockouts are assumed. However, some inventory systems may adopt a backordering policy, which violates the assumption of no stockouts. Moreover, correlated demands across items caused by cross-selling frequently occur in practice (Zhang et al., 2011). Therefore, it is more applicable to build a multi-item inventory model taking into account of correlations across items. This paper studies a multi-item inventory system comprising a major item and some minor items. Demand for the major item is independent and demand for the minor items is correlated with that for the major item. Because of cross-selling effects, if the major item is stocked out, customers will probabilistically give up buying the minor items and demand for the minor items may be pulled down to a certain degree. We assume that the unmet demand for the major item can be completely backordered, while demand for the minor items must be met without stockouts. Based on the general integer (GI) policy whereby the minor item's supply cycle is equal to a positive integer times the time length of the major item's supply cycle, we build a JRP model with complete backordering and correlated demand. Similar to the traditional JRP, the analytical solution of the proposed model cannot be found. Thus, we propose a heuristic algorithm that balances the ordering cost and the inventory holding cost by adjusting the replenishment frequencies of minor items to optimize the total inventory cost. The computational study shows that the solution attained by the heuristic is very close to the exact optimal solution. The rest of the paper is organized as follows. In Section 2, we present our assumptions and notations and then build the JRP model with complete backordering and correlated demand caused by cross-selling. The heuristic algorithm for solving the model is developed in Section 3. Section 4 presents numerical examples for computational experiments and for investigating the effectiveness of the heuristic. Section 5 concludes the paper.

If items are associated with each other because of cross-selling effects, making an inventory policy following the traditional inventory model may lead to the risk of misestimating the inventory cost. This paper presents an approach for hedging the inventory risk by considering the possible associations between items in a new JRP model with complete backordering and correlated demand. The paper makes two contributions that are different from the traditional JRP. The first is that the unmet demand for the major item can be backordered. The second is that if the major item is stocked out, the sales of minor items will be pulled down to a certain degree until the next replenishment. Through our analytical method, we observed that the optimal value of the fill rate is determined by the known parameters. Based on this fact, the model was transformed to an equivalent formulation with respect to the multiples of the major item's supply cycle. To solve the model, we proposed a heuristic algorithm based on the principle that the optimal inventory policy should balance the ordering cost and the sum of the inventory holding cost and the backordering cost. Our computational study shows that the heuristic performs well in the case where the ordering cost of the minor item is smaller than that of the major item. In the case where the ordering cost of the minor item is comparable with that of the major item, although the performance of the heuristic is not as good as that in the case of high major ordering cost, the difference is very small. In practice, the ordering cost of a minor item is often smaller than that of the major item. Thus, the proposed model and the heuristic will be applicable to real-world inventory systems. Our model holds a stringent assumption that the major item can be completely backordered, which may not be appropriate for the case of partial backordering. To make the model more practical, we need to relax this assumption and let the major item be partially backordered. Such relaxation will lead to a more intricate inventory model and some new algorithm has to be developed to obtain the optimal solution, which is the future work that we are planning to carry out.