**Inventory Models for Non-Instantaneous Deteriorating Items with Expiration Dates and Imperfect Quality under Hybrid Payment Policy in the Three-Level Supply Chain**

**Jui-Jung Liao 1, Hari Mohan Srivastava 2,3,4,5,\*, Kun-Jen Chung 6,7, Shih-Fang Lee 8, Kuo-Nan Huang 9 and Shy-Der Lin 10**


**Abstract:** This article considers an inventory model for non-instantaneous deteriorating items with expiration dates, such as seasonal items, first-hand vegetables, and fruits. Interestingly, an inspection will be performed to manage the quality of the items during the state of no deterioration because it is difficult to purchase items with 100% perfection. Additionally, we assume that the upstream member has the power of controlling or influencing downstream members' decisions. That is, the supplier asks the retailer for a partial advance paymen<sup>t</sup> to avoid cancellation of orders and offers them a credit paymen<sup>t</sup> to stimulate sales; in turn, the customer must pay some cash when placing an order and pay the remainder in credit for the retailer. The goal of this article is to determine an optimal replenishment cycle and the total annual cost function, so we explore the functional properties of the total annual cost function and show that the total annual cost function is convex. Theoretical analysis of the optimal properties shows the existence and uniqueness of the optimal solution. Then, we obtain simple and easy solution procedures for the inventory system. Moreover, numerical analysis of the inventory model is conducted, and the corresponding examples are considered with a view to illustrating the application of the supply chain model that we have investigated in this article. Finally, in the concluding section, we have not only provided the motivation and the need for our usages of mathematical analytic solution procedures based upon the convexity, monotonicity (increasing and decreasing) and differentiability properties of the object function (that is, the total annual cost function), which involve some symmetry aspects of the object function, but we have also indicated the limitations and shortcomings in our investigation, which will naturally lead to some potential directions for further research on the supply chain model, which we have considered and mathematically analyzed in this article.

**Citation:** Liao, J.-J.; Srivastava, H.M.; Chung, K.-J.; Lee, S.-F.; Huang, K.-N.; Lin, S.-D. Inventory Models for Non-Instantaneous Deteriorating Items with Expiration Dates and Imperfect Quality under Hybrid Payment Policy in the Three-Level Supply Chain. *Symmetry* **2021**, *13*, 1695. https://doi.org/10.3390/ sym13091695

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Academic Editors: Zhivorad Tomovski and Guangdong Tian

Received: 30 June 2021 Accepted: 31 August 2021 Published: 14 September 2021

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**Keywords:** inventory modeling; mathematical analytic solution procedures; economic order quantity (EOQ) model; deteriorating products; imperfect quality; hybrid paymen<sup>t</sup> policy; non-instantaneous deterioration; expiration date; trade credit financing; permissible delay in payments; object function (that is total annual cost function); supply chain managemen<sup>t</sup>
