Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 42 https://internationalpubls.com Buyers- Suppliers Win-Win Cash Flow Inventory Model with Linear Demand under Various Production Industries S. Srividhya a,*, P. Muniappan b aResearch Scholar, Department of Mathematics, Sathyabama Institute of Science And Technology, Chennai – 600 119, Tamil Nadu, India. Email: srdhasarathi17022019@gmail.com bAssistant Professor, Department of Mathematics, Sathyabama Institute of Science And Technology, Chennai – 600 119, Tamil Nadu, India. Email: munichandru@yahoo.com Article History: Received: 24-02-2024 Revised: 28-04-2024 Accepted: 09-05-2024 Abstract: This paper presents an analytical and numerical analysis of an inventory model incorporating credit periods, focusing on major business concerns that face cash flow constraints. In such circumstances, credit periods serve as a crucial tool for determining optimal production inventory strategies. The foremost objective of the research is to assess the benefits accrued by both producers and buyers within a fixed credit period framework. The analytical solution model developed in this study aims to optimize production inventory costs by considering the cycle periods and interest earned during shortage periods. Through the utilization of differential equations, the model offers a comprehensive evaluation of inventory management strategies. By incorporating credit periods as a key parameter, the research investigates how producers and buyers can derive maximum benefits from this approach. Furthermore, a numerical example is given to describe the practical application of the model and demonstrate how decision variables are employed in real-world scenarios. This example showcases the effectiveness of the analytical and numerical analysis in determining optimal production inventory costs. Overall, this study contributes to the understanding of inventory management in situations where cash flow is inconsistent. By incorporating credit periods into the analysis, the research provides insights into the decision-making process for production inventory, enabling businesses to optimize costs and enhance overall efficiency. Keywords: Permissible delay payments, Production quantity, Linear Demand, Partially backlogged shortages, time horizon. 1. Introduction Inventory management is a critical aspect for businesses across various industries and sectors. The efficient management of inventory levels is essential for meeting customer demands, minimizing costs, and maximizing profitability. One important factor that influences inventory management strategies is the consideration of credit periods in the context of production and linear demand. An inventory model that incorporates production and linear demand, along with credit periods, allows companies to make informed decisions about their inventory control strategies. By understanding the dynamics of production, demand patterns, and credit periods, businesses can optimize their inventory levels and align them with their operational and financial goals. In various fields and industries, companies face the challenge of balancing production activities with customer demand. They need to ensure that they have sufficient inventory on hand to meet customer orders promptly while avoiding excessive inventory levels that can tie up working capital. The integration of production and linear demand in an inventory model enables companies to optimize production schedules and order Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 43 https://internationalpubls.com quantities to meet demand fluctuations efficiently. Moreover, credit periods play a crucial role in managing inventory and cash flow. The availability of credit periods allows companies to delay payment for goods or services received, providing a financial advantage by freeing up working capital for other purposes. By strategically utilizing credit periods, businesses can capably manage their cash flow and strengthen their complete financial position. The integration of credit periods into the inventory model provides companies with a holistic approach to inventory management. It allows them to evaluate the trade-offs between holding costs, production costs, and financial benefits derived from credit periods. This comprehensive analysis enables businesses to make data-driven decisions on order quantities, production schedules, and payment terms. The application of the inventory model with production and linear demand using credit periods is relevant to various industries and companies. Whether in manufacturing, retail, distribution, or any other sector that deals with inventory management, the model offers a framework for optimizing inventory levels and achieving operational efficiency. By incorporating production, linear demand, and credit periods into their inventory management strategies, companies can streamline their operations, reduce costs, and enhance customer satisfaction. The utilization of this model allows businesses to strike a balance between meeting demand, minimizing holding costs, and effectively managing their cash flow. Throughout this paper, we will delve into the analytical and numerical analysis of the inventory model with production and linear demand using credit periods. We will explore the benefits and trade-offs associated with this approach and provide practical insights for companies in different fields to enhance their inventory management strategies. Through the integration of production, linear demand, and credit periods in the inventory model, companies can gain a competitive edge in their respective industries. The ability to optimize inventory levels while considering production capabilities, demand fluctuations, and financial advantages of credit periods is crucial for sustained success. In manufacturing companies, the inventory model with production and linear demand using credit periods enables efficient production planning and scheduling. By aligning production activities with anticipated demand and utilizing credit periods strategically, manufacturers can minimize stock outs, reduce production costs, and enhance customer satisfaction. The model allows for a more accurate estimation of the optimal production quantity, taking into account both the immediate demand and the anticipated future demand based on linear patterns. Retailers and distributors also benefit from the integration of credit periods into their inventory management strategies. By leveraging credit periods offered by suppliers, they can maintain optimal inventory levels without tying up excessive capital. This approach not only improves cash flow but also allows retailers to respond quickly to changing customer demands. By accurately forecasting the linear demand and factoring in credit periods, retailers can effectively determine the order quantities that will meet customer needs while managing their financial resources efficiently. Furthermore, the inventory model with production and linear demand using credit periods extends beyond manufacturing and retail sectors. It is applicable in various fields such as healthcare, e-commerce, and service-based industries. For instance, healthcare facilities can utilize the model to optimize their inventory of medical supplies, ensuring they have the right quantities available while managing costs. E- commerce companies can leverage credit periods to streamline their inventory management processes, improve fulfilment operations, and enhance customer experience. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 44 https://internationalpubls.com 1.1 LITERATURE REVIEW Zhang and Wang [1] proposed a mathematical model for inventory management in supply chain networks that takes into account multiple demand scenarios, supply chain structure, and inventory policies. The goal is to minimize the total cost of inventory while ensuring a high level of service level for customers. Zhu and Cui [2] developed a novel mathematical model for inventory control in a multi-level supply chain network. The model considers the coordination of inventory decisions between different echelons of the supply chain and aims to optimize the total inventory cost while satisfying customer demand. Li et al. [3] discussed a model for inventory management that incorporates demand uncertainty and lead time variability. The model aims to minimize the total inventory cost while ensuring a high service level for customers. Gao et al. [4] developed a stochastic mathematical model for inventory management in a multi-echelon supply chain that considers both demand uncertainty and lead time variability. The model aims to minimize the total inventory cost while ensuring a high level of service level for customers. Sarkar, B et al. [5] developed an inventory model with trade-credit policy, variable deterioration for production products. Sana, S. S [6] studied a production–inventory model for imperfect inventory process. Huang, Y. F. [7] studied an inventory model under two levels of trade credit and also limited storage space model. Hammami, R et al. [8] Analyzed carbon emissions in a multi-echelon production-stock using lead time constraints. Birim.S et al. [9] presented evaluating vendor managed inventory systems: how incentives can benefit supply chain partners. Srivathsan, S., & Kamath, M et al. [10] considered a performance modeling of a two-echelon with supply chain and information sharing. Muniappan et al. [11] developed an EOQ model for deteriorating products and time value of money for delay payments. Muniappan et al. [12] developed a production inventory model for vendor–buyer with quantity discount and backordering for used products. Mohammadi, H et al. [13] discussed about deteriorating and seasonal products, such as fresh produce, the issues of timely supply and disposal of the deteriorated products are of high concerns and presented a new mathematical model of the location-routing problem of facilities in supply chain network for deteriorating items by taking environmental reflections, cost, delivery time and customer satisfaction into account across the entire network and customer satisfaction. Amini, A et al. [14] addresses about combined transportation and inventory problem in a two-stage supply chain, including suppliers and retailers and the role of energy in terms of fuel's type selection. Vafaeenezhad et al. [15] discussed about the purchase, production and distribution quantities for facilities in a supply chain and minimizing the total cost or maximizing the profits was the major aim of supply chains, responsibility for the environmental and social impacts, processes and products, the safety and health of their employees and the entire community. Huang, J et al. [16] studied the complexities of bidders’ information interactions and behaviour preferences caused from financial and production perceptions. Also we take into account the complex and dynamic market background, which will impact the operation effect of sale policies. Yadav, A. S et al. [17] discussed about simulated Annealing to optimize FIFO & LIFO in green supply chain inventory management of Hazardous substance components industry and studied on determining the most likely level of surplus stock and shortage required for FIFO & LIFO in green supply chain stocks of Hazardous substance components industry to find the minimum total cost of the supply chain. Setak, M et al. [18] discussed two mathematical models of supply chain under Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 45 https://internationalpubls.com uncertainty. The competition is considered as a game. Davizon, Y. A et al. [19] studied the mathematical modelling, optimal control, and stability analysis for dynamic supply chain. Aghsami, A et al. [20] addresses various aspects of the blood collection centres are considered in this model and storage of optimum blood level is considered. Dehghani, E et al. [21] studied the Markov process and mixed-integer nonlinear programming is presented to design the distribution network of a supply chain. Yang et al. [22] studied the inventory competition under lead time sensitive demands and compared the consolidated scenarios. Yadav, A. S et al. [23] discussed the most likely level of surplus stock and shortage required for green supply chain stocks of Auto-components industry to find the minimum total cost in supply chain. R. Uthayakumar, & A. Ruba Priyadharshini [24] discussed a deteriorating item return policy with allowable delay and partial backlog is included in the inventory model for a single item. By selling both price and time, it optimizes the overall profit. Duary,et al. [25] A developed a very non-linear objective function for a two-warehouse inventory problem, taking into account all potential cases and subcases. In addition to applying the suggested algorithm to identify the best ideal values from an economic perspective, we employed the GRG technique to solve the problem. A two-warehouse inventory model with a decaying product whose demand varies with time, selling price, the strength of media advertisements, and continuous time with a mixed type trade credit policy is also updated here. Najafnejhad, E et al. [26] established the demand fluctuations in inventory management are greatly decreased by this policy. Based on the vendor-managed policy, this article creates an inventory model with many merchants and a single vendor. Apart from making inventory decisions, the suggested methodology maximizes an upper bound on inventory levels determined by a penalty. The established mathematical model's goal is to determine the best value for retailers' order quantities, replenishment frequencies, and upper limits on their inventory levels. Saren, S., et al. [27] demonstrates that a cap and trade policy can be used to control the overall amount of carbon released into the atmosphere by the transportation and production sectors, the lead time demand for items by retailers is assumed to be random rather than fixed uniform and normal distribution functions. The ideal retailer lot size, customer service rendered by the store, and retailer reorder points are evaluated under these two distribution functions. Ganguly,et al. [28] examined a reworking approach that would be put into practice following an inventory of such faulty goods and there would be no shortages when the assembled product was remanufactured. Productivity variation was raised to enhance the quality of the completed products while lowering manufacturing costs. The space capacity and budget were regarded as limitations. Based on five distinct distribution functions and the defined variable parameters of production rate, manufacturing batch size, and backorder amount, the total inventory cost was computed. Mondal, et al. [29] discussed a three-tiered supply chain management model based on a single manufacturer, one supplier, and several retailers, all subject to payment and advertising regulations. Advertising has shown to have a beneficial impact on sales since it creates demand for the goods in the market. By taking into account a single-setup multiple-delivery policy, variable transportation costs, variable carbon emissions costs, and trade-credit policy, the model seeks to minimize supply chain costs and maximize profit. Based on the payment duration, the objective function is optimized for certain instances. Sen, N et al. [30] establishes a green supply chain model with one supplier and one buyer for decaying commodities. Demand is influenced by the selling price and the degree of greening Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 46 https://internationalpubls.com improvement. There are two established generalized models for green supply chains: one with consignment stock policy and the other without. 1.2 Research Focus The integration of production, linear demand, and credit periods in the inventory model offers significant benefits for companies across diverse fields. By optimizing inventory levels, aligning production with demand, and strategically utilizing credit periods, businesses can achieve operational efficiency, cost savings, and improved financial performance. The analytical and numerical analysis of this inventory model provides valuable insights and practical guidance for companies to make informed decisions regarding their inventory management strategies. Throughout this paper, we will delve further into the specific methodologies, analyses, and numerical examples that illustrate the advantages of the inventory model with production and linear demand using credit periods. We aim to provide readers with a comprehensive understanding of how this model can be applied in various industries and companies, ultimately enabling them to optimize their inventory management practices and achieve their business objectives. 1.3 Decision variables The permissible delay 𝑀 Time horizon 𝑇, 𝑇1 Optimum total inventory cost 𝑇𝐶(𝑇1, 𝑇) Backlogged shortage 𝛿 2. FORMULATION AND ANALYTICAL SOLUTION FOR INVENTORY MODEL In the proposed inventory model, the change in inventory over a [0, T1) time interval is determined by the production and the linear demand. Additionally, the occurrence of shortages in subsequent time intervals is taken into account. The model formulation can be described as follows the inventory level at the beginning of the time interval as 𝐼1(𝑡), the production quantity during the time interval [0, T1)as P. The demand rate (linear) during the time interval [0, T1) as D(t). The shortage quantity during the time interval [T1, T) as S(t). Hence, the change in inventory over the time interval [0, T) can be calculated as: 𝑑𝐼1(𝑡) 𝑑𝑡 + 𝜃𝐼1(𝑇1) = 𝑃 − 𝑎 − 𝑏𝑡 ; [0, T1). (1) 𝐼1(𝑇1) = 0 gives, 𝐼1(𝑡) = 𝜃(𝑃−𝑎)+𝑏 𝜃2 [1 − 𝑒𝜃(𝑇1−𝑡)] + 𝑏 𝜃 [𝑇1𝑒𝜃(𝑇1−𝑡) − 𝑡] 𝑑𝐼2(𝑡) 𝑑𝑡 + 𝜃𝐼2(𝑇1) = − 𝑎 1+ 𝛿 (𝑇−𝑡) ; [T1, T). ( 2) 𝐼2(𝑇1) = 0 gives, 𝐼2(𝑡) = − 𝑎 𝛿 [𝑙𝑜𝑔(1 + 𝛿(𝑇 − 𝑇1)) − 𝑙𝑜𝑔(1 + 𝛿(𝑇 − 𝑡)) ] We obtain inventory level for the different intervals as follows: Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 47 https://internationalpubls.com 𝐼1(𝑡) = 𝜃(𝑃−𝑎)+𝑏 𝜃2 [1 − 𝑒𝜃(𝑇1−𝑡)] + 𝑏 𝜃 [𝑇1𝑒𝜃(𝑇1−𝑡) − 𝑡] (3) 𝐼2(𝑡) = − 𝑎 𝛿 [𝑙𝑜𝑔(1 + 𝛿(𝑇 − 𝑇1)) − 𝑙𝑜𝑔(1 + 𝛿(𝑇 − 𝑡)) ] (4) I (T) = 𝐼1(𝑡) + 𝐼2(𝑡); [0, T) (5) Various Inventory cost calculated by using integral calculus in the interval [0, T1) We are using holding cost ℎ, shortage cost 𝑠, Deterioration cost 𝑝, Opportunity cost 𝛼. HC = ℎ ∫ 𝐼1(𝑡)𝑑𝑡 𝑇1 0 = ℎ { 𝜃(𝑃−𝑎)+𝑏 𝜃3 [1 − 𝑒𝜃𝑇1] + 𝑇1 𝜃2 (𝜃(𝑃 − 𝑎) + 𝑏 (𝑒𝜃𝑇1 − 𝜃𝑇1 2 )) } (6) DC = 𝑝𝜃 ∫ 𝐼1(𝑡)𝑑𝑡 𝑇1 0 = 𝑝𝜃 { 𝜃(𝑃−𝑎)+𝑏 𝜃3 [1 − 𝑒𝜃𝑇1] + 𝑇1 𝜃2 (𝜃(𝑃 − 𝑎) + 𝑏 (𝑒𝜃𝑇1 − 𝜃𝑇1 2 )) } (7) SC = 𝑠 ∫ 𝐼2(𝑡)𝑑𝑡 𝑇 𝑇1 = 𝑠𝑎 𝛿2 [𝛿(𝑇 − 𝑇1) − log (1 + 𝛿(𝑇 − 𝑇1))] (8) OC = 𝛼 ∫ 𝐼2(𝑡)𝑑𝑡 𝑇 𝑇1 = 𝑎𝛼 𝛿 {𝛿(𝑇 − 𝑇1) − log (1 + 𝛿(𝑇 − 𝑇1)) } (9) 2.1. Scenario1: supplier’s delay payment period 𝑀 ≤ 𝑇1 Consider a situation where a buyer purchases goods or services from a supplier on credit, with specific payment terms and conditions. In this case, the supplier's permissible delay, denoted as M, represents the maximum duration allowed for the buyer to settle the outstanding payment. Let's say the total credit period agreed upon between the supplier and the buyer is denoted as 𝑇1. It is specified that within each credit cycle, the buyer earns interest during the interval [0, 𝑇1), referred to as IE1, at a rate denoted as Ie. However, in the subsequent interval [M, 𝑇1), denoted as Ip, the buyer is required to pay interest to the supplier at a predetermined interest rate, denoted as Ir. To provide a real-life example, let's consider a retailer purchasing goods from a wholesaler on credit terms. The wholesaler allows a credit period of 60 days (𝑇1) for the retailer to make the payment. However, the retailer has the flexibility to delay payment for up to 30 days (M) before it is considered overdue. During the first 30 days of the credit period ([0, 30), denoted as IE1), the retailer benefits from earning interest on the outstanding amount owed to the wholesaler. However, if the retailer delays the payment beyond the permissible delay (M) and falls within the interval [30, 60) days (denoted as Ip), it is required to pay interest to the wholesaler at the predetermined interest rate, Ir. This serves as a mechanism to incentivize the buyer to make timely payments and compensate the supplier for the delayed payment. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 48 https://internationalpubls.com In summary, this real-life example illustrates a scenario where a buyer earns interest during the initial portion of the credit period [0, 𝑇1) (denoted as IE1), but once the permissible delay (M) is exceeded, the buyer is obliged to pay interest to the supplier during the subsequent period [M, 𝑇1) (denoted as Ip). We have, IE1 = 𝑝𝐼𝑒 ∫ (𝑇1 − 𝑡)(𝑎 + 𝑏𝑡)𝑑𝑡 𝑇1 0 = 𝑝𝐼𝑒 𝑇1 2 6 [3𝑎 + 𝑏𝑇1] (10) Ip = 𝑝𝐼𝑟 ∫ 𝐼(𝑡)𝑑𝑡 𝑇1 𝑀 = 𝑝𝐼𝑟 { 1−𝑒𝜃(𝑇1−𝑀) 𝜃3 [𝜃(𝑃 − 𝑎) + 𝑏(1 − 𝜃𝑇1)] + 𝜃(𝑃−𝑎)+𝑏 𝜃2 [𝑇1 − 𝑀] − 𝑏 2𝜃 (𝑇1 2 − 𝑀2)} (11) PC = ∫ (𝑎 + 𝑏𝑡)𝑐𝑑𝑑𝑡 𝑇1 0 = 𝑐𝑑 [𝑎𝑇1 + 𝑏𝑇1 2 2 ] (12) Average cost developed as follow TC1 = 1 𝑇 {𝑟 + ℎ { 𝜃(𝑃−𝑎)+𝑏 𝜃3 [1 − 𝑒𝜃𝑇1] + 𝑇1 𝜃2 (𝜃(𝑃 − 𝑎) + 𝑏 (𝑒𝜃𝑇1 − 𝜃𝑇1 2 ))} + 𝑃𝜃 { 𝜃(𝑃−𝑎)+𝑏 𝜃3 [1 − 𝑒𝜃𝑇1] + 𝑇1 𝜃2 (𝜃(𝑃 − 𝑎) + 𝑏 (𝑒𝜃𝑇1 − 𝜃𝑇1 2 ))} + 𝑠 { 𝑎 𝛿2 [𝛿(𝑇 − 𝑇1) − log (1 + 𝛿(𝑇 − 𝑇1))]} + 𝑎𝛼 𝛿 {𝛿(𝑇 − 𝑇1) − log (1 + 𝛿(𝑇 − 𝑇1)) } + 𝑃𝐼𝑟 { 1−𝑒𝜃(𝑇1−𝑀) 𝜃3 [𝜃(𝑃 − 𝑎) + 𝑏(1 − 𝜃𝑇1)] + 𝜃(𝑃−𝑎)+𝑏 𝜃2 [𝑇1 − 𝑀] − 𝑏 2𝜃 (𝑇1 2 − 𝑀2)} +𝑐𝑑 [𝑎𝑇1 + 𝑏𝑇1 2 2 ] − 𝑝𝐼𝑒 𝑇1 2 6 [3𝑎 + 𝑏𝑇1]} = 1 𝑇 {𝑟 + (ℎ + 𝑃𝜃) { 𝜃(𝑃−𝑎)+𝑏 𝜃3 [1 − 𝑒𝜃𝑇1] + 𝑇1 𝜃2 (𝜃(𝑃 − 𝑎) + 𝑏 (𝑒𝜃𝑇1 − 𝜃𝑇1 2 )) } + 𝑎(𝑠+𝛿𝛼) 𝛿2 {𝛿(𝑇 − 𝑇1) − log (1 + 𝛿(𝑇 − 𝑇1)) } + 𝑃𝐼𝑟 { 1−𝑒𝜃(𝑇1−𝑀) 𝜃3 [𝜃(𝑃 − 𝑎) + 𝑏(1 − 𝜃𝑇1)] + 𝜃(𝑃−𝑎)+𝑏 𝜃2 [𝑇1 − 𝑀] − 𝑏 2𝜃 (𝑇1 2 − 𝑀2)} +𝑐𝑑 [𝑎𝑇1 + 𝑏𝑇1 2 2 ] − 𝑝𝐼𝑒 𝑇1 2 6 [3𝑎 + 𝑏𝑇1]} (13) 𝜕𝑇𝐶1(𝑇1,𝑇) 𝜕𝑇1 = 0 𝑎𝑛𝑑 𝜕𝑇𝐶1(𝑇1,𝑇) 𝜕𝑇 = 0 2.1.1. Solution procedure for optimum inventory level [ 𝜕2𝑇𝐶1(𝑇1,𝑇) 𝜕𝑇1 2 ] 𝑎𝑡 (𝑇1 ∗ ,𝑇∗) > 0 , [ 𝜕2𝑇𝐶1(𝑇1,𝑇) 𝜕𝑇2 ] 𝑎𝑡 (𝑇1 ∗ ,𝑇∗) > 0 ; [( 𝜕2𝑇𝐶1(𝑇1,𝑇) 𝜕𝑇1 2 ) ( 𝜕2𝑇𝐶1(𝑇1,𝑇) 𝜕𝑇2 ) − ( 𝜕2𝑇𝐶1(𝑇1,𝑇) 𝜕𝑇1𝜕𝑇 ) 2 ] > 0 𝜕𝑇𝐶1(𝑇1,𝑇) 𝜕𝑇1 = 0 𝑎𝑛𝑑 𝜕𝑇𝐶1(𝑇1,𝑇) 𝜕𝑇 = 0 Implies the optimal values of 𝑇∗ and 𝑇1 ∗ Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 49 https://internationalpubls.com 𝜕𝑇𝐶1(𝑇1, 𝑇) 𝜕𝑇1 = 0 1 𝑇 {(ℎ + 𝑃𝜃) {(𝜃(𝑃 − 𝑎) + 𝑏) ( −𝑒𝜃𝑇1 𝜃2 ) + 𝑇1 𝜃 [𝑏 (𝑒𝜃𝑇1 − 1 2 )] + 1 𝜃2 [(𝜃(𝑃 − 𝑎) + 𝑏 (𝑒𝜃𝑇1 − 𝜃𝑇1 2 ))] } − 𝑎(𝑠+𝛿𝛼)(𝑇−𝑇1) 1+𝛿(𝑇−𝑇1) + 𝑃𝐼𝑟 {– 𝑏 ( 1−𝑒𝜃(𝑇1−𝑀) 𝜃2 ) − 𝑒𝜃(𝑇1−𝑀) 𝜃2 [𝜃(𝑃 − 𝑎) + 𝑏(1 − 𝜃𝑇1)] + 𝜃(𝑃−𝑎)+𝑏 𝜃2 − 𝑏𝑇1 𝜃 } +𝑐𝑑[𝑎 + 𝑏𝑇1] − 𝑝𝐼𝑒 𝑇1 3 [3𝑎 + 𝑏𝑇1] − 𝑝𝐼𝑒 𝑏𝑇1 2 6 } = 0 (14) 𝜕𝑇𝐶1(𝑇1, 𝑇) 𝜕𝑇 = 0 1 𝑇 { 𝑎(𝑠+𝛿∝)(𝑇−𝑇1) 1+𝛿(𝑇−𝑇1) } − 1 𝑇2 {𝑟 + (ℎ + 𝑃𝜃) { 𝜃(𝑃−𝑎)+𝑏 𝜃3 [1 − 𝑒𝜃𝑇1] + 𝑇1 𝜃2 (𝜃(𝑃 − 𝑎) + 𝑏 (𝑒𝜃𝑇1 − 𝜃𝑇1 2 )) } + 𝑎(𝑠+𝛿𝛼) 𝛿2 {𝛿(𝑇 − 𝑇1) − log (1 + 𝛿(𝑇 − 𝑇1)) } + 𝑃𝐼𝑟 { 1−𝑒𝜃(𝑇1−𝑀) 𝜃3 [𝜃(𝑃 − 𝑎) + 𝑏(1 − 𝜃𝑇1)] + 𝜃(𝑃−𝑎)+𝑏 𝜃2 [𝑇1 − 𝑀] − 𝑏 2𝜃 (𝑇1 2 − 𝑀2)} +𝑐𝑑 [𝑎𝑇1 + 𝑏𝑇1 2 2 ] − 𝑝𝐼𝑒 𝑇1 2 6 [3𝑎 + 𝑏𝑇1]} = 0 (15) Hence the optimum cycle length values are 𝑇1 ∗, 𝑇∗ and optimum average inventory cost is 𝑇𝐶1(𝑇1, 𝑇) 2.2. Scenario2: supplier’s payment delay 𝑻𝟏 < 𝑀 Consignment inventory refers to a situation where the supplier stocks and maintains inventory at the buyer's premises, but the buyer does not pay for the inventory until it is consumed or sold. In this scenario, the buyer benefits from having access to the inventory without incurring any immediate costs. Consider a manufacturing company (buyer) that relies on a supplier for raw materials. The supplier agrees to provide consignment inventory to the buyer, ensuring a constant supply of raw materials. The agreement specifies that within the time interval [0, M), the buyer earns interest on the inventory held and does not pay any interest or carrying costs to the supplier during this period. During this time interval, the buyer can utilize the consignment inventory to fulfil production demands without paying for the materials upfront. This arrangement allows the buyer to effectively manage cash flow by deferring the payment until the inventory is consumed or transformed into finished goods. Meanwhile, the buyer can invest the funds that would have been allocated for purchasing inventory elsewhere, potentially earning interest on those funds during the consignment period. Overall, the consignment inventory arrangement exemplifies a real-life scenario where the buyer benefits from earning interest on inventory held within the specified time interval [0, M), without incurring any interest or carrying costs associated with the supplier's consigned inventory. Therefore we have, IE2 = 𝑝𝐼𝑒 {∫ (𝑇1 − 𝑡)(𝑎 + 𝑏𝑡)𝑑𝑡 + (𝑀 − 𝑇1 ) ∫ (𝑎 + 𝑏𝑡)𝑑𝑡 𝑇1 0 𝑇1 0 } = 𝑝𝐼𝑒 𝑇1 2 6 [3𝑎 + 𝑏𝑇1] + 𝑝𝐼𝑒 𝑇1(𝑀−𝑇1) 2 [2𝑎 + 𝑏𝑇1] (16) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 50 https://internationalpubls.com The total average cost developed as TC2 = 𝑟 + 𝐻𝐶 + 𝐷𝐶 + 𝑆𝐶 + 𝑂𝐶+𝑃𝐶− 𝐼𝐸2 𝑇 (17) = 1 𝑇 {𝑟 + ℎ { 𝜃(𝑃−𝑎)+𝑏 𝜃3 [1 − 𝑒𝜃𝑇1] + 𝑇1 𝜃2 (𝜃(𝑃 − 𝑎) + 𝑏 (𝑒𝜃𝑇1 − 𝜃𝑇1 2 ))} + 𝑃𝜃 { 𝜃(𝑃−𝑎)+𝑏 𝜃3 [1 − 𝑒𝜃𝑇1] + 𝑇1 𝜃2 (𝜃(𝑃 − 𝑎) + 𝑏 (𝑒𝜃𝑇1 − 𝜃𝑇1 2 ))} + 𝑠 { 𝑎 𝛿2 [𝛿(𝑇 − 𝑇1) − log (1 + 𝛿(𝑇 − 𝑇1))]} + 𝑎𝛼 𝛿 {𝛿(𝑇 − 𝑇1) − log (1 + 𝛿(𝑇 − 𝑇1)) } +𝑐𝑑 [𝑎𝑇1 + 𝑏𝑇1 2 2 ] − 𝑃𝐼𝑒 𝑇1 2 6 [3𝑎 + 𝑏𝑇1] − 𝑝𝐼𝑒 𝑇1(𝑀−𝑇1) 2 [2𝑎 + 𝑏𝑇1]} = 1 𝑇 {𝑟 + (ℎ + 𝑃𝜃) { 𝜃(𝑃−𝑎)+𝑏 𝜃3 [1 − 𝑒𝜃𝑇1] + 𝑇1 𝜃2 (𝜃(𝑃 − 𝑎) + 𝑏 (𝑒𝜃𝑇1 − 𝜃𝑇1 2 )) } + 𝑎(𝑠+𝛿𝛼) 𝛿2 {𝛿(𝑇 − 𝑇1) − log (1 + 𝛿(𝑇 − 𝑇1)) }+𝑐𝑑 [𝑎𝑇1 + 𝑏𝑇1 2 2 ] − 𝑝𝐼𝑒 𝑇1 2 6 [3𝑎 + 𝑏𝑇1] − 𝑝𝐼𝑒 𝑇1(𝑀−𝑇1) 2 [2𝑎 + 𝑏𝑇1]} (18) 2.2.1. Solution procedure for optimum inventory level To solve 𝜕𝑇𝐶2(𝑇1,𝑇) 𝜕𝑇1 = 0 𝑎𝑛𝑑 𝜕𝑇𝐶2(𝑇1,𝑇) 𝜕𝑇 = 0 and the sufficient conditions are [ 𝜕2𝑇𝐶2(𝑇1,𝑇) 𝜕𝑇1 2 ] 𝑎𝑡 (𝑇1 ∗ ,𝑇∗) > 0 , [ 𝜕2𝑇𝐶2(𝑇1,𝑇) 𝜕𝑇2 ] 𝑎𝑡 (𝑇1 ∗ ,𝑇∗) > 0 𝑎𝑛𝑑 [( 𝜕2𝑇𝐶2(𝑇1,𝑇) 𝜕𝑇1 2 ) ( 𝜕2𝑇𝐶2(𝑇1,𝑇) 𝜕𝑇2 ) − ( 𝜕2𝑇𝐶2(𝑇1,𝑇) 𝜕𝑇1𝜕𝑇 ) 2 ] > 0 1 𝑇 {(ℎ + 𝑃𝜃) {(𝜃(𝑃 − 𝑎) + 𝑏) ( −𝑒𝜃𝑇1 𝜃2 ) + 𝑇1 𝜃 [𝑏 (𝑒𝜃𝑇1 − 1 2 )] + 1 𝜃2 [(𝜃(𝑃 − 𝑎) + 𝑏 (𝑒𝜃𝑇1 − 𝜃𝑇1 2 ))] } − 𝑎(𝑠+𝛿𝛼)(𝑇−𝑇1) 1+𝛿(𝑇−𝑇1) +𝑐𝑑[𝑎 + 𝑏𝑇1] − 𝑝𝐼𝑒 (𝑎 + 𝑏𝑇1)(𝑀 − 𝑇1)} = 0 (19) 𝜕𝑇𝐶2(𝑇1,𝑇) 𝜕𝑇 = 0 1 𝑇 { 𝑎(𝑠+𝛿𝛼)(𝑇−𝑇1) 1+𝛿(𝑇−𝑇1) } − 1 𝑇2 {𝑟 + (ℎ + 𝑃𝜃) { 𝜃(𝑃−𝑎)+𝑏 𝜃3 [1 − 𝑒𝜃𝑇1] + 𝑇1 𝜃2 (𝜃(𝑃 − 𝑎) + 𝑏 (𝑒𝜃𝑇1 − 𝜃𝑇1 2 )) } + 𝑎(𝑠+𝛿𝛼) 𝛿2 {𝛿(𝑇 − 𝑇1) − log (1 + 𝛿(𝑇 − 𝑇1)) }+𝑐𝑑 [𝑎𝑇1 + 𝑏𝑇1 2 2 ] − 𝑝𝐼𝑒 𝑇1 2 6 [3𝑎 + 𝑏𝑇1] − 𝑝𝐼𝑒 𝑇1(𝑀−𝑇1) 2 [2𝑎 + 𝑏𝑇1]} (20) Hence the optimal values are 𝑇1 ∗ and 𝑇∗ and optimum average cost is 𝑇𝐶2(𝑇1 ∗, 𝑇∗). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 51 https://internationalpubls.com 3. NUMERICAL ANALYSIS Table 3.1 Changes in different decision variables 𝜹 M 2 3 4 5 TC (T1, T) 𝑻𝟏 ∗ 𝑻∗ 4.4963 X 1028 827.1952 826.6952 9.2133 X 1026 778.6209 778.2875 1.8418 X 1025 729.7397 729.4897 10 TC (T1, T) 𝑻𝟏 ∗ 𝑻∗ 3.0291 X 1021 620.8900 620.3900 2.2013 X 1017 501.8787 501.5454 1.3824 X 1013 381.1116 380.8616 15 TC (T1, T) 𝑻𝟏 ∗ 𝑻∗ 1.8893 X 1014 413.7335 413.2335 7.8069 X 107 230.4927 230.1594 7.1454 X 104 50.2504 49.9543 20 TC (T1, T) 𝑻𝟏 ∗ 𝑻∗ 1.0331 X 107 205.0625 204.5625 5.3775 X 104 36.8832 36.4306 7.5777 X 103 168.5181 168.2681 25 TC (T1, T) 𝑻𝟏 ∗ 𝑻∗ 1.3602 X 107 5.9916 6.2550 7.2276 X 106 200.5001 200.1668 3.1132 X 104 506.2087 505.9587 Table 3.2 Changes in different decision variables 𝜹 M 1 2 3 4 5 5 TC (T1,T) 𝑻𝟏 ∗ 𝑻∗ 4.1905 X 1043 1.4135 X 103 1.4125 X 103 1.1096 X 1038 1.2301 X 103 1.2296 X 103 6.2022 X 1033 1.0903 X 103 1.0899 X 103 1.5567 X 1030 971.8674 971.6174 9.3828 X 1026 866.0034 865.8034 10 TC (T1,T) 𝑻𝟏 ∗ 𝑻∗ 2.4093 X 1035 1.1425 X 103 1.1415 X 103 3.7857 X 1027 885.9160 885.4160 1.1996 X 1021 672.2785 675.9451 1.5939 X 1015 479.2169 478.9669 4.4645 X 109 297.0809 296.8809 30 TC (T1,T) 𝑻𝟏 ∗ 𝑻∗ 5.1465 X 1035 30.4084 29.3617 3.6045 X 1028 173.2545 172.7545 7.5704 X 1021 698.5739 698.2406 2.4025 X 1036 1.1754 X 103 1.1751 X 103 2.4001 X 1049 1.6009 X 103 1.6007 X 103 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 52 https://internationalpubls.com Table 3.3 Changes in different decision variables 𝜹 M 1 2 3 5 TC (T1, T) 𝑻𝟏 ∗ 𝑻∗ 3.1050 X 105 34.9379 34.8524 9.2133 X 1026 778.6209 778.2875 1.8418 X 1025 729.7397 729.4897 10 TC (T1, T) 𝑻𝟏 ∗ 𝑻∗ 2.4945 X 105 27.8632 26.7505 1.7809 X 105 19.9712 19.4182 2.6640 X 104 13.8226 13.4452 15 TC (T1, T) 𝑻𝟏 ∗ 𝑻∗ 1.8503 X 105 20.3688 19.2046 9.6210 X 104 10.3770 9.7606 1.8597 X 104 1.2340 1.7273 25 TC (T1, T) 𝑻𝟏 ∗ 𝑻∗ 2.2438 X 104 4.1077 0.0161 8.4983 X 104 9.1159 8.4776 1.7930 X 105 20.3517 19.9900 35 TC (T1, T) 𝑻𝟏 ∗ 𝑻∗ 9.8379 X 104 10.3926 8.9951 2.4016 X 105 27.3753 26.8382 3.1132 X 1017 33.3314 32.9819 45 TC (T1, T) 𝑻𝟏 ∗ 𝑻∗ 2.3268 X 105 26.1037 24.9813 3.5302 X 105 40.5510 40.0281 3.2294 X 105 37.2245 36.8770 55 TC (T1, T) 𝑻𝟏 ∗ 𝑻∗ 3.5347 X 105 40.1939 39.1227 4.2710 X 105 49.1483 48.6314 2.9820 X 105 34.4051 34.0561 Table 3.4 Changes in different decision variables M 𝜹 5 15 1 TC (T1, T) 𝑻𝟏 ∗ 𝑻∗ 4.0237 X 1030 883.3441 882.3441 1.1256 X 1023 666.0408 665.0408 2 TC (T1, T) 𝑻𝟏 ∗ 𝑻∗ 2.7562 X 1028 821.0799 820.5799 8.9539 X 1018 548.1489 547.6489 3 TC (T1, T) 𝑻𝟏 ∗ 𝑻∗ 6.5556 X 1026 774.3687 774.0354 2.3585 X 1015 445.2460 444.9126 4 TC (T1, T) 𝑻𝟏 ∗ 𝑻∗ 3.0652 X 1025 736.1033 735.8533 1.2527 X 1012 351.1609 350.9109 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 53 https://internationalpubls.com 5 TC (T1, T) 𝑻𝟏 ∗ 𝑻∗ 2.1144 X 1024 702.6979 702.4979 1.0747 X 109 263.1678 262.9678 10 TC (T1, T) 𝑻𝟏 ∗ 𝑻∗ 2.3901 X 1019 560.4200 560.3200 8.7991 X 103 2.7822 2.8177 20 TC (T1, T) 𝑻𝟏 ∗ 𝑻∗ 3.6397 X 109 278.3660 278.3160 1.8342 X 103 1.0174 X 103 0.0174 X 103 25 TC (T1, T) 𝑻𝟏 ∗ 𝑻∗ 2.4325 X 105 139.7781 139.7381 1.0987 X 103 2.1046 X 103 2.1046 X 103 4. CONCLUSION The two scenarios presented in the paper shed light on different dynamics in the buyer-supplier relationship regarding permissible delay and interest rates. The findings highlight the importance of understanding and optimizing credit terms in inventory management strategies. 4.1 Results and Discussion 4.1.1. Scenario 1: Supplier's Permissible Delay M ≤ T1 In this scenario, where the supplier's permissible delay M is less than or equal to the total credit period T1, the analysis reveals a distinct pattern in interest earnings and payments for the buyer. During each credit cycle, the buyer earns interest (IE1) in the interval [0, T1) while paying interest (Ip) in the interval [M, T1). This model encourages the buyer to settle the payment within the permissible delay to avoid additional costs. The implications of this scenario suggest that the buyer can benefit from earning interest on the outstanding amount during the initial period of the credit term. However, if the buyer delays payment beyond the permissible delay, the buyer incurs interest charges, serving as a financial incentive for timely payment. This scenario emphasizes the importance of managing cash flow effectively to maximize interest earnings and minimize interest payments. 4.1.2. Scenario 2: Supplier's Permissible Delay T1 < M In this alternative scenario, the supplier's permissible delay T1 is less than the total credit period M. Notably, during the interval [0, M), denoted as IE2, the buyer earns interest at the Ie rate without paying any interest to the supplier. The discussion surrounding this scenario reveals a different dynamic in the buyer-supplier relationship. The buyer has the advantage of earning interest on the outstanding amount during the entire permissible delay period [0, M) without incurring any interest payments to the supplier. This situation presents an opportunity for the buyer to utilize the available cash resources strategically, potentially generating additional income through interest-earning investments. However, it is crucial for the buyer to manage the payment effectively and settle the outstanding amount within the permissible delay (M) to avoid additional interest charges. The findings suggest that the buyer's ability to optimize cash flow and leverage the interest-earning potential during the permissible delay can significantly impact the overall financial performance. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 54 https://internationalpubls.com Overall, both scenarios emphasize the importance of effectively managing credit terms, permissible delays, and interest rates in inventory management. The findings provide valuable insights into the dynamics of the buyer-supplier relationship, enabling businesses to make informed decisions and optimize their inventory strategies based on the specific credit terms and financial incentives involved. Furthermore, the analysis of these two scenarios underscores the significance of aligning credit terms with the cash flow dynamics and financial goals of the buyer. The choice of permissible delay and the corresponding interest rates can have a substantial impact on the buyer's ability to earn interest, manage costs, and optimize working capital. In Scenario 1, where the permissible delay is within or equal to the total credit period, the buyer faces the risk of incurring interest payments if the payment is delayed beyond the permissible limit. This setup encourages prompt payment and serves as a mechanism to incentivize the buyer to maintain a healthy cash flow and minimize financial costs. It highlights the importance of effective cash flow management and timely payment to leverage interest earnings while avoiding additional expenses. On the other hand, Scenario 2 presents a scenario where the permissible delay exceeds the total credit period. In this case, the buyer has the advantage of earning interest throughout the entire permissible delay without incurring any interest payments to the supplier. This scenario provides the buyer with an opportunity to strategically utilize cash resources and potentially earn additional income through interest-earning investments. However, it also emphasizes the importance of disciplined financial management to ensure timely payment within the permissible delay and avoid any negative consequences such as interest charges. Overall, the analysis of these scenarios emphasizes the need for businesses to carefully consider and negotiate credit terms with suppliers to align with their financial objectives. Optimizing credit terms can lead to improved cash flow, reduced financial costs, and enhanced profitability. By understanding the dynamics of permissible delay and interest rates, buyers can make informed decisions regarding inventory management, working capital allocation, and financial strategies. It is important to note that the specific implications and outcomes of these scenarios may vary depending on the industry, market conditions, and individual buyer-supplier relationships. Therefore, businesses should conduct a thorough analysis of their unique circumstances and consider the potential trade-offs and benefits associated with different credit terms and interest rate structures. 4.2 Research contribution The objective of the developed analytical solution model is to determine the optimal production inventory cost by considering the cycle period and incorporating the interest earned during shortage periods. Through the model, the study aims to identify the most cost-effective production and inventory strategies. To illustrate the effectiveness of the model, a numerical example is provided to showcase the optimal values of the decision variables. This example demonstrates the practical application of the analytical solution in real-world scenarios and highlights the potential cost savings and efficiency improvements achievable through its implementation. Furthermore, the model has the potential for extension to accommodate additional factors such as price breaks and various holding costs. By incorporating these elements, the model can offer a more comprehensive and accurate representation of the inventory management problem, enabling businesses to make informed decisions and optimize their operations based on specific pricing and holding cost considerations. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 55 https://internationalpubls.com Overall, the developed analytical solution model provides a valuable framework for evaluating and optimizing production inventory costs. Its flexibility to incorporate various factors and its potential for extension make it a versatile tool for businesses seeking to enhance their inventory management strategies and achieve cost savings. REFERENCES [1] Zhang, H., & Wang, Y. (2021). A mathematical model for inventory management in supply chain networks. Journal of Industrial Engineering and Management, 14(2), 321-333. [2] Zhu, Y., & Cui, Y. (2021). A novel mathematical model for inventory control in a multi-level supply chain network. Applied Mathematical Modelling, 88, 79-94. [3] Li, Q., Li, X., & Zhou, Y. (2021). A new mathematical model for inventory management in a two-echelon supply chain. Journal of Intelligent Manufacturing, 32(1), 181-194. [4] Gao, J., Li, X., & Li, Y. (2021). A stochastic mathematical model for inventory management in a multi-echelon supply chain. International Journal of Production Economics, 231, 107903. [5] Sarkar, B., Saren, S., & Cárdenas-Barrón, L. E. (2015). An inventory model with trade-credit policy and variable deterioration for fixed lifetime products. Annals of Operations Research, 229(1), 677-702. [6] Sana, S. S. (2010). A production–inventory model in an imperfect production process. European Journal of Operational Research, 200(2), 451-464. [7] Huang, Y. F. (2006). An inventory model under two levels of trade credit and limited storage space derived without derivatives. Applied Mathematical Modelling, 30(5), 418-436. [8] Hammami, R., Nouira, I., & Frein, Y. (2015). Carbon emissions in a multi-echelon production-inventory model with lead time constraints. International Journal of Production Economics, 164, 292-307. [9] Birim, S., & Sofyalioglu, C. (2017). Evaluating vendor managed inventory systems: how incentives can benefit supply chain partners. Journal of Business Economics and Management, 18(1), 163-179. [10] Srivathsan, S., & Kamath, M. (2017). Performance modeling of a two-echelon supply chain under different levels of upstream inventory information sharing. Computers & Operations Research, 77, 210-225. [11] Muniappan, P., Uthayakumar, R., & Ganesh, S. (2015). An EOQ model for deteriorating items with inflation and time value of money considering time-dependent deteriorating rate and delay payments. Systems Science & Control Engineering, 3(1), 427-434. [12] Muniappan, P., Uthayakumar, R., & Ganesh, S. (2016). A production inventory model for vendor–buyer coordination with quantity discount, backordering and rework for fixed life time products. Journal of Industrial and Production Engineering, 33(6), 355-362. [13] Mohammadi, H., &EhteshamRasi, R. (2022). Multi-Objective Mathematical Model for Locating Flow Optimization Facilities in Supply Chain of Deteriorating Products. Journal of System Management, 8(1), 51-71. [14] Amini, A., &Ghodsi, R. (2016). A linear mathematical model for a transportation-inventory problem in a two-stage supply chain with different types of fuels for vehicles. International Journal of Services and Operations Management, 25(3), 347-360. [15] Vafaeenezhad, T., Tavakkoli-Moghaddam, R., &Cheikhrouhou, N. (2019). Multi-objective mathematical modeling for sustainable supply chain management in the paper industry. Computers & Industrial Engineering, 135, 1092- 1102. [16] Huang, J., & Song, J. (2018). Optimal inventory control with sequential online auction in agriculture supply chain: An agent-based simulation optimisation approach. International Journal of Production Research, 56(6), 2322-2338. [17] Yadav, A. S., Abid, M. O. H. A. M. M. E. D., Bansal, S. H. I. K. H. A., Tyagi, S. L., & Kumar, T. A. N. U. J. (2020). FIFO & LIFO in green supply chain inventory model of hazardous substance components industry with storage using simulated annealing. Advances in Mathematics: Scientific Journal, 9(7), 5127-5132. [18] Setak, M., Feizizadeh, F., Tikani, H., &Ardakani, E. S. (2019). A bi-level stochastic optimization model for reliable supply chain in competitive environments: Hybridizing exact method and genetic algorithm. Applied Mathematical Modelling, 75, 310-332. [19] Davizon, Y. A. (2020). Mathematical Modeling, Optimal Control and Stability Analysis for Dynamic Supply Chains. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2s (2024) 56 https://internationalpubls.com [20] Aghsami, A., Abazari, S. R., Bakhshi, A., Yazdani, M. A., Jolai, S., &Jolai, F. (2023). A meta-heuristic optimization for a novel mathematical model for minimizing costs and maximizing donor satisfaction in blood supply chains with finite capacity queueing systems. Healthcare Analytics, 100136. [21] Dehghani, E., Pishvaee, M. S., &Jabalameli, M. S. (2018). A hybrid Markov process-mathematical programming approach for joint location-inventory problem under supply disruptions. RAIRO-Operations Research, 52(4-5), 1147-1173. [22] Yang, J. Q., Zhang, X. M., Fu, H. Y., & Liu, C. (2017). Inventory competition in a dual-channel supply chain with delivery lead time consideration. Applied Mathematical Modelling, 42, 675-692. [23] Yadav, A. S., Kumar, A. M. I. T., Agarwal, P. R. I. Y. A. N. K. A., Kumar, T. A. N. U. J., &Vanaja, R. (2020). LIFO in green supply chain inventory model of auto-components industry with warehouses using differential evolution. Advances in Mathematics: Scientific Journal, 9(7), 5121-5126. [24] R. Uthayakumar, & A. Ruba Priyadharshini. (2024). Optimal strategy on inventory model under permissible delay in payments and return policy for deteriorating items with shortages. Malaya Journal of Matematik, 12(01), 71–84. https://doi.org/10.26637/mjm1201/006 [25] Duary, A., Das, S., Arif, M. G., Abualnaja, K. M., Khan, M. A. A., Zakarya, M., & Shaikh, A. A. (2022). Advance and delay in payments with the price-discount inventory model for deteriorating items under capacity constraint and partially backlogged shortages. Alexandria Engineering Journal, 61(2), 1735-1745. [26] Najafnejhad, E., Tavassoli Roodsari, M., Sepahrom, S., & Jenabzadeh, M. (2021). A mathematical inventory model for a single-vendor multi-retailer supply chain based on the Vendor Management Inventory Policy. International Journal of System Assurance Engineering and Management, 12(3), 579-586. [27] Saren, S., Guchhait, R., AlArjani, A., & Sarkar, B. (2023). Developing trust among players in a vendor-managed inventory model for random demand under environmental impact. Mathematical Biosciences and Engineering, 20(9), 16169-16193. [28] Ganguly, B., Dey, B. K., Pareek, S., & Sarkar, B. (2023). Cost-Effective Imperfect Production-Inventory System under Variable Production Rate and Remanufacturing. Mathematics, 11(15), 3417. [29] Mondal, A. K., Pareek, S., & Sarkar, B. (2024). Payment policy for a three-echelon supply chain management under advertisement-driven demand. RAIRO-Operations Research, 58(1), 45-77. [30] Sen, N., Bardhan, S., & Giri, B. C. (2024). Consignment based integrated inventory model for deteriorating goods with price-and green-sensitive demand. Sādhanā, 49(1), 1-17.