مدل موجودی فروشنده خریدار یکپارچه با تخفیف قیمت سفارش معوقه و سرمایه گذاری موثر برای کاهش هزینه های سفارش

The single-vendor single-buyer integrated production inventory system has been an object of study for a long time, but little is known about the effect of investing in reducing ordering cost on the integrated inventory models with backorder price discount and variable lead time. The purpose of this article is to investigate in the continuous review model with backorder price discount and variable lead time to effectively increase investment and to reduce the joint expected annual total cost. The integrated strategy discussed here is one in which the buyer orders a quantity, then the vendor produces n times order quantity in each production cycle, in order to reduce setup cost. In addition, the buyer offers backorder price discounts to the customers that may motivate the customers’ desire for backorders, and buyer ordering cost can be reduced through effective investment. An integrated inventory model is established to find the optimal solutions of order quantity, ordering cost, backorder price discount, lead time, and the number of shipments from the vendor to the buyer in one production run, so that the joint expected annual total cost incurred has the minimum value. Furthermore, numerical examples are used to demonstrate the benefits of the model.

The integrated inventory management system is a common practice in the global markets and provides economic advantages for both the vendor and the buyer. In recent years, most integrated inventory management systems have focused on the integration between vendor and buyer. Once they form a strategic alliance in order to minimize their own cost or maximize their own profit, then trading parties can collaborate and share information to achieve improved benefits. Therefore, several authors (e.g., Amasaka, 2002, Ben-Daya and Hariga, 2004, Bylka, 2003, Chang et al., 2006, Hoque and Goyal, 2006, Ouyang et al., 2007a, Pan and Hsiao, 2005, Villa, 2001, Viswanathan, 1998, Yang and Wee, 2001 and Zhang et al., 2007) have presented the integrated inventory management system. The integration between vendor and buyer for improving the performance of inventory system control has been discussed for years. Goyal (1976) is among the first who analyzed an integrated inventory model for a single-vendor single-buyer system. The framework he proposed has encouraged many researchers to present various types of integrated inventory system. Banerjee (1986) modified Goyal’s (1976) model and presented a joint economic-lot-size model where a vendor produces for a buyer to order on a lot-for-lot basis. Goyal (1988) further generalized Banerjee’s (1986) model by relaxing the assumption of the lot-for-lot policy of the vendor and suggested that the vendor’s economic production quantity should be a positive integer multiple of the buyer’s purchase quantity. Ha and Kim (1997) further generalized Goyal’s (1988) model and presented an integrated lot-splitting model of facilitating multiple shipment in small lots. Hill (1999) proposed a more general batching and shipping policy involving the successive shipment size of the first m shipments increases by a fixed factor and remaining shipments would be equal sized. In a recently study, Pan and Yang (2002) generalized Goyal’s (1988) model by considering lead time as a decision variable and obtained a lower joint total expected cost and shorter lead time. Yang and Pan (2004) considered variable lead time and quantity improvement investment with normal distributional demand in the model proposed in Pan and Yang, 2002 and Ouyang et al., 2004 extend Pan and Yang (2002) and developed a single-vendor single-buyer integrated production inventory model under the assumption that the lead time demand is stochastic and lead time is decision variable. All the aforementioned integrated vendor–buyer inventory systems treat the ordering cost and/or lead time as constants. However, in the practical market, ordering cost and lead time can be controlled and reduced in various ways. For example, lead time can be reduced at an added crashing cost; ordering cost reduction can be attained through worker training, procedural changes, and specialized equipment acquisition; in other words, the lead time is controllable, and the ordering cost can be reduced through further investment. It has been a trend by shortening the lead time and reducing ordering cost; we can lower the safety stock, reduce the stockout loss, and improve the service level to the customer so as to increase the competitive edge in business. On the other hand, in the real market, as unsatisfied demands occur, we can often observe that some customers may prefer their demands to be backordered, and some may refuse the backorder case. When a shortage occurs, many factors may affect the customers’ willingness of accepting backorders. For example, for well-famed products or fashionable goods such as certain brand gum shoes, hi-fi equipment, cosmetics, and clothes, customers may prefer to wait for backorders. Hence, how to motivate the customers to wait for backorders is a valuable problem. This means that we should endeavor to generate high customer loyalty so that the customers would like to accept backorders. The factor is an offering of a price discount from the buyer to customers (see, Chuang et al., 2004, Ouyang et al., 2007b and Pan and Hsiao, 2001). In general, provided that a buyer could offer a price discount on the stockout item by negotiation to secure more backorders, it may make the customers more willing to wait for the desired items. Through controlling a price discount, we could generate high customer loyalty. This means that we could reduce cost of lost-sales and reduce holding cost, and then minimize the relevant inventory total cost. For example, Procter & Gamble, Southwest Airlines, Nike, Disney, Nordstrom, Wal-Mart, McDonald’s, Marriott Hotels, and several Japanese (Sony, Toyota, Canon) and European (IKEA, Club Med, Bang & Olufsen, Electrolux, Nokia, Lego, Tesco) companies. These companies focus on the customer and are organized to respond effectively to changing customer needs. Undoubtedly, these companies endeavor to generate high customer loyalty, so that, by price discount, they can raise the customer’s incentive to wait for backorder (see Kotler and Keller (2006, chap. 2)). In other words, the bigger the discount, the bigger the advantage to the customers, and hence, a larger number of backorder ratio may result. This phenomenon reveals that, as unsatisfied demands occur during the stockout period, how to find an optimal backorder ratio through controlling a price discount from a buyer to minimize the relevant inventory total cost is a decision-making problem worth discussing.

The primary purpose of this paper is to present the single-vendor single-buyer integrated production inventory model with backorder price discount and effectively increase investment to reduce the ordering cost. We present two centralized models of commonly used investment cost function, logarithmic and power, to be employed for ordering cost reduction and offered backorder price discount to customers that may motivate the customers’ desire for backorders. By analyzing the joint expected annual total cost, we developed an algorithm to determine the optimal order quantity, ordering cost, backorder price discount, lead time, and the numbers of shipments per production run from the vendor to the buyer. The results of the numerical examples indicate that if the buyer makes his/her decisions with the capital investment in reducing ordering cost and offering backorder price discount to customers, it will help to lower the system cost, and we can obtain a significant amount of savings to increase the competitive edge in business. In future researches on this problem, on the one hand we would consider deteriorating items in this model and on the other since this paper limits lead time to be deterministic; it would be interesting to consider the procurement lead time as a random variable and discussing the effects in reducing lead time variability. Another possible extension of this work may be conducted by considering the vendor’s provision of a permissible delay in payments in this integrated inventory model.