سیاست های تعیین اندازه دسته تولید برای موارد رو به وخامت گذاشته با تاریخ انقضا و اعتبار تجاری جزئی به مشتریان ریسک اعتباری

In practice, a credit-worthy retailer frequently receives a permissible delay on the entire purchase amount without collateral deposits from his/her supplier (i.e., an up-stream full trade credit). By contrast, a retailer usually requests his/her credit-risk customers to pay a fraction of the purchase amount at the time of placing an order, and then grants a permissible delay on the remaining balance (i.e., a down-stream partial trade credit). In addition, many products such as blood banks, pharmaceuticals, fruits, vegetables, volatile liquids, and others deteriorate constantly and have their expiration dates. However, not many researchers have taken the expiration date of a deteriorating item into consideration. The purpose of this paper is to establish optimal lot-sizing policies for a retailer who sells a deteriorating item to credit-risk customers by offering partial trade credit to reduce his/her risk. The proposed model is a generalized case of many previous models. By applying theorems in pseudo-convex fractional functions, we can easily prove that the optimal solution not only exists but is also unique. Moreover, we propose three discrimination terms, which can easily identify the optimal solution among all possible alternatives. Finally, some numerical examples are presented to highlight the theoretical results and managerial insights.

Harris (1913) established the classical economic order quantity (EOQ) model based on the assumptions that the purchase items are non-perishable and can be sold indefinitely, and the retailer must pay for the entire purchase cost as soon as the purchase items are received. In reality, many products (e.g., fruits, vegetables, medicines, volatile liquids, blood banks and others) not only deteriorate continuously (e.g., evaporation, obsolescence, and spoilage) but also have their expiration dates. Ghare and Schrader (1963) derived a revised form of the economic order quantity (hereafter EOQ) model by assuming exponential decay. Then Covert and Philip (1973) extended Ghare and Schrader׳s constant deterioration rate to a two-parameter Weibull distribution. Dave and Patel (1981) considered an EOQ model for deteriorating items with time-proportional demand when shortages were prohibited. Sachan (1984) further extended the model to allow for shortages. Goswami and Chaudhuri (1991) generalized an EOQ model for deteriorating items from a constant demand pattern to a linear trend in demand. Hariga (1996) established optimal EOQ models for deteriorating items with time-varying demand. Goyal and Giri (2001) provided a survey on the recent trends in modeling of deteriorating inventory. Skouri et al. (2009) considered inventory models with ramp-type demand rate and Weibull deterioration rate. Skouri et al. (2011) further generalized the model to add a permissible delay in payments under consideration. Dye (2013) provided some results on finding the optimal replenishment and preservation technology strategies for a non-instantaneous deteriorating inventory model. Recently, Chen et al. (2013b) proposed economic production quantity (EPQ) models for deteriorating items. All the above mentioned papers did not consider the fact that deteriorating items have their expiration dates. In fact, the study of deteriorating items with expiration dates has received a relatively little attention in the literature. Currently, Bakker et al. (2012) provided an excellent review of inventory systems with deterioration since 2001. In practice, a seller frequently offers his/her buyers a permissible delay in payment (i.e., trade credit) for settling the purchase amount. Usually, there is no interest charge if the outstanding amount is paid within the permissible delay period. However, if the payment is not paid in full by the end of the permissible delay period, then interest is charged on the outstanding amount. Goyal (1985) proposed an EOQ model under conditions of permissible delay in payments. Aggarwal and Jaggi (1995) extended Goyal׳s model for deteriorating items. Jamal et al. (1997) further generalized Aggarwal and Jaggi׳s model to allow for shortages. Chang et al. (2003) developed an EOQ model for deteriorating items under supplier credits linked to ordering quantity. Huang (2003) proposed an EOQ model in which the supplier offers the retailer a permissible delay and the retailer in turn provides his/her customers another permissible delay to stimulate demand. Ouyang et al. (2006) established an EOQ model for deteriorating items to allow for partial backlogging under trade credit financing. Liao (2007) presented an EPQ model for deteriorating items under permissible delay in payments. Teng (2009) developed ordering policies for a retailer who offers distinct trade credits to its good and bad credit customers. Hu and Liu (2010) presented an EPQ model with permissible delay in payments and allowable shortages. Teng et al. (2011) extended an EOQ model for stock-dependent demand to supplier׳s trade credit with a progressive payment scheme. Teng et al. (2012) generalized traditional constant demand to non-decreasing demand. Lou and Wang (2013) proposed an integrated inventory model with trade credit financing in which the vendor decides his/her production lot size while the buyer determines his/her expenditure. Lately, Chen et al. (2013a) built up an EOQ model when conditionally permissible delay links to order quantity. Most of the above mentioned articles assumed that buyers are good-credit customers and receive full trade credit. Hence, the use of other trade credit strategies to reduce default risks with credit-risk customers has received a relatively little attention in the literature. Recently, Seifert et al. (2013) presented an excellent review of trade credit financing. Some relevantly recent articles in trade credit financing were developed by Chern et al. (2013), Taleizadeh (2014), Wu et al. (2014), and Yang et al. (2013). In this paper, an EOQ model for deteriorating items with expiration dates was developed in a supply chain with up-stream full trade credit and down-stream partial trade credit financing. The proposed model is a generalized case of Goyal (1985), Teng (2002), Huang (2003), Teng and Goyal (2007), Teng (2009), and Chen and Teng, 2014 and Chen and Teng, 2014. Further, by applying the existing theoretical results in pseudo-convex fractional functions, we can easily prove that the optimal solution not only exists but also is unique. Moreover, we propose three discrimination terms to identify the optimal solution among possible alternatives. Finally, some numerical examples are presented to highlight the theoretical results and managerial insights.

The use of a down-stream partial trade credit to reduce default risks with credit-risk customers has received a very little attention by the researchers. In this paper, we have built an EOQ model for deteriorating items with maximum lifetime in a supply chain in which the retailer receives an up-stream full trade credit from his/her supplier while offers a down-stream partial trade credit to his/her credit-risk customers. The proposed model has been a generalized model including Goyal (1985), Teng (2002), Huang (2003), Teng and Goyal (2007), Teng (2009), and Chen and Teng (2014) as special cases. By applying the theoretical results in convex fractional programs, we have obtained the necessary and sufficient conditions for finding a unique optimal solution. Furthermore, we have proposed three discrimination terms to identify the optimal solution among alternatives. Finally, we have used several numerical examples to show all possible alternatives. Following most traditional EOQ models, we have adopted the objective to minimize the total relevant cost. However, one of two anonymous referees suggests an excellent alternative in which the objective is to maximize the total profit by assuming that the ending inventory is not zero and can be sold at salvage value in order to reduce accelerating deterioration cost as time approaches to the expiration date. As a result, one may extend our EOQ model from zero ending-inventory to non-zero ending-inventory. Also, one could generalize the model to allow for shortages and partial backlogging. Finally, the proposed model with single player can be extended to an integrated cooperative model for both players (e.g., the supplier and the retailer) or a non-cooperative Nash or Stackelberg equilibrium solution for each player.