**1. Introduction**

In today's competitive business environment, more than 80% of businesses offer their products on various short-term, interest-free credit terms (that is, a credit payment) with a view to stimulate sales and to reduce inventory in the United Kingdom and the United States of America. Likewise, trade credit financing is used by approximately 60% of international trade transactions, rendering it to be the second after that of banks and other financial institutions in the United States of America. Additionally, in order to avoid the risk of order cancellation and non-paymen<sup>t</sup> risks, the business frequently offers a partial credit period to the downstream members, who must pay a portion of the procurement amount at the time of placing an order and then receive a permissible delay on the rest of the outstanding amount (that is, a cash-credit payment). On the other hand, granting trade credit increases not only the opportunity loss, but also the default risk from the viewpoint of the business, so the powerful businesses may ask the downstream members to prepay the entire or a fraction of the procurement amount before the delivery to mitigate interest loss and default risks (that is, an advance payment). For example, insurance companies generally require an advance paymen<sup>t</sup> in order to extend coverage to the insured party.

In existing literatures, Zhang [1] proposed an advance paymen<sup>t</sup> plan because it may save time and money for a customer to prepay, for example 80.00 USD for 4 months of water bills, instead of paying 20.00 USD each month for 4 months. All the above mentioned paymen<sup>t</sup> types can be combined such that, for example, the supplier demands the retailer to prepay 5 to 10% of the total procurement cost as a good-faith deposit when both sign a contract of agreemen<sup>t</sup> to install some item(s). Upon delivery of the item(s), a cashon-delivery paymen<sup>t</sup> to cover the supplier's cost of the item(s) is then required. In this contractual arrangement, the retailer will pay the remainder of the total cost after the work is completed.

In reality, the deteriorating items have a maximum lifetime due to their physical nature and must be disposed of after the expiration date, due to the fact that consumers evaluate the freshness of a deteriorating item by checking its expiration date before making a purchase, and the willingness to purchase a deteriorating item decreases throughout its shelf-life. Furthermore, the expiration date is the most important factor that is time-bound and plays an important role in developing the inventory model. In practice, most of the products maintain their quality or original situation over a span; that is, during this span, deterioration does not occur, and then they begin to deteriorate in the next period. It is observed that foodstuffs, first-hand vegetables, and fruits have a short span during which fresh quality is maintained and there is almost no spoilage. These processes are defined as the non-instantaneous deterioration of the product.

The quality of the products is considered to be another direct factor to affect a consumer's purchase decision as well. Furthermore, this article assumes that the retailer receives the items with a time-varying deterioration rate depending on its expiration date, such as seasonal products, and that an inspection will occur during the state of no deterioration in order to manage the quality of the products. By performing the screening process, the retailer detects the imperfect items and throws them out.

In this article, we first establish an inventory model for non-instantaneous deteriorating items with expiration dates and imperfect quality in which we assume a 100% screening process to identify imperfect items. We then consider that the supplier asks the retailer to prepay a fraction of the procurement cost when signing a contract to buy products, to pay another fraction of the procurement cost in cash upon receiving the ordered quantity, and

to receive a short-term interest-free credit term on the remaining procurement cost (that is, an advance-cash-credit payment). Likewise, the retailer gives the customer the opportunity to pay a fraction of the procurement cost after delivery of the ordered items and then to pay the remaining procurement cost at a later date without any additional charges to reduce the risk of cancellations of the order from customers (that is, a cash-credit payment). It is worth mentioning that, by the usage of the mathematical analytic solution procedures, the present article shows that the total annual cost function is convex by exploring the functional properties of the total annual cost function such as, for example, the continuity, convexity, monotonicity (increasing and decreasing), and differentiability properties, whereby one can also see the symmetry aspects of the total annual const function. Furthermore, by applying the mathematical analytic solution procedures again, we prove that the retailer's optimal replenishment cycle not only exists, but it also is unique. With a view to illustrating and validating the proposed inventory model, we have considered numerical examples involving different fixed markup rates. Finally, in the concluding section of this article (Section 6), we have briefly discussed the limitations and shortcomings of this investigation in that we have concentrated upon the inventory system without shortage and that it can affect the supply chain from the producer to the retailer. Furthermore, this model has the potential to be extended to incorporate inflation and quantity discount effects, different demand forms such as credit-linked promotion-dependent demand and other issues under the system with shortages. Additionally, this article has considered the deterministic situation, so considering the stochastic situation, such as stochastic demand, can be another future research direction on the subject of this article.
