مدیریت موجودی فروشنده زنجیره تامین با وخامت مواد اولیه و محصولات

Fast deteriorating raw materials such as raw milk, fruit and vegetables are commonly used to produce slowly deteriorating finished products such as milk powders, cheeses, and pastas. This paper studies a Vendor Managed Inventory (VMI) type supply chain where the manufacturing vendor decides how to manage the system-wide inventories of its fast deteriorating raw material and its slowly deteriorating product. The decision variables are a common replenishment cycle of the product and the replenishment frequency of the raw material. We assume the deteriorating rates are known constants and every retailer's demand is deterministic. We develop an integrated model to calculate the total inventory and deterioration cost for such a system. We prove the convexity of the cost functions, and based on this a golden search algorithm is developed to find the optimal solution of the model. Our numerical results show that the deteriorating rate of the product may increase the total cost by more than 40% compared to the zero-deteriorating rate, while the deteriorating raw material has less impact on the total cost (commonly less than 5% in our numerical examples). This indicates that more attention should be paid to the product than the raw material. Further, an increase in the number of retailers can make the replenishment frequency of the raw material increase significantly but the common replenishment cycle of the product decreases a little. This indicates that adding a new retailer would not be felt strongly by the other retailers but would be felt by the supplier of the raw material.

A long evolution of supply chain cooperation led to the emergence of Vendor Managed Inventory (VMI) type supply chains in the 1980s. A VMI system suggests that the vendor manages the inventories of its own and its multiple downstream retailers. The inventory management of one vendor and “m” retailers is centralized to reduce total supply chain inventory costs. In practice, VMI partnerships have increased inventory turnovers of Wal-Mart and Procter & Gamble (P&G) by 30% ( Buzzell and Ortmeyer, 1995), and have contributed to Barilla gaining a competitive edge as the largest pasta vendor in the world, with 35% of the pasta sold in Italy and 22% of the pasta sold in Europe ( Hammond, 2003). Food vendors, such as Barilla from Italy, Kraft Foods from the United States, and McCain Foods from Canada, constantly deal with deteriorating items. One of the greatest advantages of implementing VMI is to provide their retailers and their end customers with foods (like cheeses and pastas) as fresh as possible. For example, by adopting VMI, Barilla uses a common replenishment cycle policy to replenish its retailers. This policy suggests that: (1) Every retailer is replenished once in a common replenishment cycle. For example, the cycle is 3 days for three retailers, and each retailer can be replenished on one of these 3 days (see also Yu et al., 2009a, Yu et al., 2009b and Zhang et al., 2007). (2) Every retailer's inventory is replenished immediately after its product is produced by the manufacturer to the amount meeting the requirement of the retailer in the common replenishment cycle. The redundant stock of deteriorating foods is then prevented. Moreover, the raw materials such as raw milk and strawberries of these manufacturers (vendors) deteriorate quickly and have also to be replenished to the manufacturer in a high frequency from agricultural producers. To reduce deterioration, the raw materials are commonly replenished several times in every common replenishment cycle of the finished product. Therefore, many food manufacturers as the vendors in VMI systems face the following critical question: What is the optimal common replenishment cycle of its product and how often should the raw material be replenished in a common replenishment cycle so as to minimize the system-wide inventory and deterioration costs? To the best of our knowledge, the above question challenging food manufacturers is still left unanswered in the literature (for the details refer to Section 2). This paper aims to answer this question by focusing on a VMI system consisting of a food manufacturer and m retailers. The manufacturer determines the common replenishment cycle of the product and the number of order times of the raw material in a common cycle to produce the product. The product is produced by the manufacturer at a fixed production rate, and is replenished to its retailers without delay in accordance with the above VMI policy. The demand of each retailer is deterministic. The contribution of this paper is two-fold. It is one of the first to study a food-industry VMI system where a common replenishment policy is used to manage the inventories of a fast deteriorating raw material and a slow deteriorating product. An integrated model to obtain the total inventory and deterioration costs is given in a closed form. Next, we prove the convexity of the cost function in the common replenishment cycle, and develop an exact algorithm to find the optimal solution of the model in seconds. The application of the algorithm is demonstrated with numerical examples. The rest of this paper is organized as follows. Section 2 reviews the related literature. Section 3 describes the research problem and lists all required notations used throughout the paper. In Section 4, we analyze the VMI system and develop an integrated VMI model considering a deteriorating product and a raw material. Section 5 develops a corresponding algorithm to find the optimal solution of the integrated model. Section 6 conducts a detailed numerical study on the application of the algorithm and analyzes the influence of deterioration rates of the product and the raw material on the total cost. Section 7 concludes the paper.

This is one of the first papers focusing on an inventory problem for a VMI system where both the raw material and the finished product are subject to deterioration. The vendor, as a manufacturer, buys a deteriorating raw material to produce a finished product at a finite production rate and transports it to its retailers. The problem of determining the optimal common replenishment cycle for the finished product and the replenishment frequency for the raw material is modeled as an integrated model to minimize the system-wide inventory and deterioration costs. We develop an algorithm to find the optimal solutions. From our models and numerical examples, we obtain the following managerial insights: • The deteriorating finished product should be managed carefully because it can influence the total cost significantly, compared to a non-deteriorating product. The total cost can increase by more than 40% due to deterioration (see Table 3). This explains why a company often uses frozen or sealed products to replace fresh products, like pastas, because the deterioration cost for a fresh product is very high. • Although the deterioration rate of the raw material is faster than that of the finished product, managers should attach more importance to the latter. This is because deterioration for the finished product happens at all the vendor's and retailers' warehouses (see Fig. 1), but happens only once for the raw material at the vendor's warehouse. For example, for pizzas made of vegetables, our results indicate that for the pizza vendor the deterioration of the pizzas is more important in the total VMI cost than the deterioration of the vegetable raw materials. The algorithm to find an optimal policy is given in Section 5.2. • When a manager intends to deviate away from the optimal solution, the common replenishment cycle of the finished product and the replenishment frequency of the raw material should be changed in the same direction (simultaneously increase or decrease) to avoid a large increase in the total cost. As illustrated in Fig. 2, the total cost is less sensitive to the common replenishment cycle of the finished product and the replenishment frequency of the raw material in one diagonal direction of a simultaneous increase than in the other diagonal direction. This is applicable to different combinations of deterioration rates of the raw material and finished product, such as from grapes to wines and from vegetables to frozen pastas. • With an increase in the number of retailers, the optimal common replenishment cycle is little changed and the change is very much less than that of the replenishment frequency of the raw material when the number of retailers is not small (e.g. no less than 5 in Table 5). In practice, this indicates that when a new retailer is added into the VMI system, the other retailers (e.g. supermarkets) may not experience much change in the product replenishment cycle, but the supplier of the raw material would be requested to replenish the raw material more frequently. This goes some way to explaining why the replenishment cycle of a deteriorating product (e.g., milk) at various different supermarkets is often quite stable. • Counter-intuitively, a negative correlation between the optimal common replenishment cycle and the optimal replenishment frequency of the raw material in the cycle is found in the case where the number of retailers changes. When the number of retailers increases, the common replenishment cycle decreases a little since the cycle is shared with all the other retailers (i.e., determined by the total demand of all retailers). However, the frequency of the raw material replenishment often needs to increase for meeting the demand of an added retailer. This suggests managers that the positive correlation between the optimal common replenishment cycle and the optimal replenishment frequency of the raw material in other cases (e.g. in Fig. 2) should not be taken for granted in the case where the number of retailers changes.