مقدار سفارش اقتصادی و قیمت خرید برای اقلام با کیفیت بازرسی ناقص هنگامی که از خریدار به تامین کننده منتقل می شود

Traditional Economic Order Quantity (EOQ) models, implicitly assume that all items that are received are perfect. Although recent EOQ models for items with imperfect quality, which relax this assumption, are closer to real-world problems, they implicitly assume that suppliers do not conduct a full inspection. In this paper, we study the relationship between buyer and supplier with regard to conducting the inspection and resulting in a change the buyer’s economic order quantity and purchasing price. We model and analyze the problem under two conditions: (1) assuming there is no relationship between the buyer’s selling price, buyer’s purchasing price, and customer demand; (2) assuming there is relationship between the buyer’s selling price, buyer’s purchasing price, and customer demand. Numerical examples are provided to illustrate the models.

Traditional economic order quantity (EOQ) models offer a mathematical approach to determine the optimal number of items a buyer should order to a supplier each time. One major implicit assumption of these models is that all the items are of perfect quality. However, this is not always the case, as in some situations a percentage of the items is imperfect. Porteus (1986) and Rosenblatt and Lee (1986) were among the first to study the effect of imperfect items on EOQ and EPQ (economic production quantity) models. Following them, several studies have incorporated the effect of imperfect items into different EOQ and EPQ models, most notably Salameh and Jaber (2000), who considered a situation in which an average p% of all items ordered are imperfect. The buyer conducts an inspection of all the items to separate the imperfect items from the perfect ones, after which the imperfect items are assumed to be sold as a single batch at the end of inspection process. Formulating the problem, the optimal order quantity is derived. Salameh and Jaber's (2000) model has been extended in several directions. For example Goyal and Cárdenas-Barrón (2002) corrected a small error in the original model. Rezaei (2005), Papachristos and Konstantaras (2006), Wee et al. (2007) and Chang and Ho (2010) studied the problem by considering the occurrence of shortage. Chung and Huang (2006) identified the optimal order quantity with imperfect items for retailers when delay in payment is allowed. Hsu and Yu (2009) formulated the problem to determine the optimal order quantity under a one-off discount. Chan et al. (2003) divided imperfect items into three categories: imperfect items that can be sold at a lower price, imperfect items that can be reworked and imperfect items that should be rejected, after which they devised a mathematical model to determine the EPQ. Wang (2005) proposed a mathematical model to determine the EPQ and optimal inspection. Haji and Haji (2010) formulated and solved the problem in situations where imperfect items are reworked with a random rate. In a recent study, Maddah and Jaber (2008) corrected some of the flaws of the original study by Salameh and Jaber (2000), which, although mathematically interesting, led to no significant changes in the final results. Some researchers considered imperfect items in the context of buyer-supplier relationships. Huang, 2002 and Huang, 2004 and Goyal et al. (2003), for example, formulated models to determine the optimal integrated buyer-supplier inventory policy for imperfect items and found that joint decision-making can reduce the expected annual cost of inventory significantly. Chen and Kang, 2007 and Chen and Kang, 2010 formulated the problem after considering the delay in payment. They assumed that the supplier can increase the warranty cost to maintain the long-term relationship. Rezaei and Davoodi (2008) formulated a model to determine the optimal lot-size including imperfect items and select the suppliers simultaneously, while Lin (2009) formulated a model for a single-supplier/single-buyer relationship to determine the optimal lot-size when some items are imperfect. Quality of received items has been also recognized in the literature of supplier selection as one of the most important criteria (see for example Rezaei and Davoodi, 2008 and Rezaei and Davoodi, 2011). For a detailed review and discussion of the extensions of EOQ model for imperfect quality items, see Khan et al. (2011). A review of existing literature reveals that several extensions of the problem have been proposed. However, in all the relevant studies, the inspection process is assumed to be conducted by the buyer. In this paper, we study the problem from a different perspective and propose a suitable framework to determine how the buyer can help the supplier carry out the inspection and reduce the imperfect rate. The rest of this paper is organized as follows. Section 2 contains the mathematical modeling of the problem under two different conditions. In Section 3, numerical examples are presented and Section 4, finally, contains the conclusion and suggestions for future research.

In this paper, we have formulated and solved a problem to determine the maximum purchasing price a buyer is willing to pay to a supplier to avoid receiving imperfect items under two conditions. First, we assumed that the buyer's selling price is independent of the buyer's purchasing price, while under the second condition, we assumed that changing the buyer's purchasing price influences the buyer's selling price and customer demand. We solved a number of numerical examples, the results of which show that, in both cases, in addition to the purchasing price, the buyer agrees to pay more than the screening cost to avoid receiving imperfect items. Paying this additional amount implies that the supplier conducts the inspection process. We have also shown how the buyer can help the supplier to improve production quality by paying some more than the usual purchasing price. It is clear that the screening process costs the supplier much less than the buyer, as the supplier is more familiar with the product and its deficiencies. Therefore the results of this paper show that having the inspection process conducted by the buyer is no longer cost-efficient. For future research, we suggest studying situations where buyer and supplier conduct the inspection together. While existing literature focuses on imperfection due to production quality, incorporating other causes of imperfection into the problem, such as transition, is suggested. We also suggest taking the findings of this study into account when formulating the supplier selection problems. Finally, we believe that the proposed model in this paper can be extended to be used in the context of supplier development, where the main concern is improving the supplier’s performance and capabilities. That is, while this paper considers the reasonability of improving the quality of supplied items, studying the improvement of other features of the items (e.g., delivery) as a way to develop suppliers may be considered as well.