Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 255 https://internationalpubls.com Two Warehouse Inventory Model for Deteriorating Items with Trade Credit Under Inflation Monika Rania, Sachin Kumarb , Vikas Tiwaric, Vipin Kumard* a. Research Scholar, Department of Mathematics, AKTU, Lucknow, U.P., India. b. Department of Applied Sciences, KIET Group of Institutions Delhi - NCR, Ghaziabad c. Department of Applied Sciences, Rajkiya Engineering College, Sonbhadra, UP, India d. Department of Mathematics, B.K. Birla Institute of Engineering and Technology, Pilani Raj. India (Corresponding author) (drvkmaths@gmail.com ) Article History: Received: 14-11-2024 Revised:26-12-2024 Accepted:10-01-2025 Abstract: Establishing warehouses is essential in societies where commercial activities have expanded significantly, making efficient storage a prerequisite for smooth exchange. Variable holding costs play a crucial role in determining warehouse expenses, as these costs typically increase over time. This study examines an inventory model for perishable goods stored in two types of warehouses (rented and owned) under conditions of partial backlogging and inflation. Along with exploring sustainable marketing strategies, the article investigates the connections between product pricing, advertising, and demand, highlight the considerable impact of price reductions and promotions by advertisement. Additionally, this study addresses a trade credit scheme where the supplier grants the retailer a fixed time frame to finalize the account. The research aims to enhance operational efficiency and reduce overall costs by optimizing key decision factors such as storage capacity, credit period, and replenishment time period. To explore the effects on the system’s optimal total cost, along with managerial insights, numerical examples and sensitivity analyses are provided. The model’s outcomes are validated through sensitivity analysis using Mathematica 13.0 software, ensuring their robustness and reliability. Keywords: Inventory Model, Two-Warehouse, Deteriorating Items, Trade Credit, Inflation 1. Introduction The establishment of efficient and strategically located warehouses is a fundamental necessity in modern commercial societies, where the rapid expansion of trade and commerce demands robust storage solution policies. Warehouses serve as pivotal nodes in the supply chain, facilitating the smooth exchange of goods between suppliers, retailers, and consumers. A significant factor in the management of warehouses is the variable holding cost, which represents the expenses incurred for storing goods over time. These costs are not static; they tend to escalate due to factors such as spoilage, obsolescence, and inflation. Particularly in the context of perishable goods, managing holding costs becomes even more critical, as these goods have a limited shelf life and are favorable to rapid depreciation. Thus, understanding and controlling these costs is crucial for maintaining the warehouse operations. This study focuses on developing and analyzing an inventory model specifically designed for perishable goods. The model considers two types of warehouses: rented and owned. Each type of mailto:drvkmaths@gmail.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 256 https://internationalpubls.com warehouse comes with its unique set of cost structures. Rented warehouses often involve variable rental costs and less control over the infrastructure, while owned warehouses require significant capital investment and incur maintenance expenses. The model addresses these differences and aims to provide a balanced approach to managing inventory across both types of storage facilities. In addition to the core focus on warehouse management, the study explores sustainable marketing strategies that can influence the demand for perishable goods. The relationships between advertising, product pricing, and consumer demand are complex and impactful. Effective promotional activities and strategic price reductions can significantly boost demand, thereby reducing the time goods spend in storage and mitigating holding costs. The research emphasizes the importance of aligning marketing efforts with inventory management practices to achieve optimal results. Another critical aspect introduced in this study is the trade credit policy, where suppliers grant retailers a specific period to settle their accounts. This policy is a vital point of financial management within the supply chain, which provides the flexibility to the retailers to manage the cash flows and optimize their inventory levels. By extending the payment period, retailers can invest in larger inventories without immediate financial strain, with the better bulk purchasing rates and improved inventory turnover. The aim of this research is to enhance operational efficiency and reduce overall costs by optimizing key factors such as the storage capacity, credit period, and replenishment time. By these parameters, businesses can achieve a more cost-effective and responsive inventory management system, particularly suited to the challenges associated with perishable goods. In summary, this study provides a comprehensive framework for managing perishable goods in a dual- warehouse environment. It highlights the significance of variable holding costs, the strategic role of advertising and pricing in demand management, and the benefits of flexible trade credit policies. The findings offer valuable insights and practical solutions for businesses seeking to enhance their operational frameworks, reduce costs, and improve their market responsiveness and sustainability. Through this research, we aim to contribute to the ongoing efforts in optimizing supply chain management and promoting more efficient commercial practices. 2. Literature Review: We have presented an inventory model for deteriorating items under the effect of inflation along with two warehouse system and trade credit policy. Here in this section literature review is provided with Two warehouse system, deterioration, Trade credit period and inflation keywords. Two Warehouse: The inclusion of two types of warehouses—rented and owned—in an inventory model for perishable goods is a strategic decision driven by various operational and financial considerations. The necessity of incorporating both rented and owned warehouses arises from the need to balance cost efficiency, flexibility, and risk management in inventory management. Jaggi et al. (2015) examined the effect of deterioration along with imperfect quality on an inventory model of two- warehouse. Tiwari et al. (2016) analyzed a retailer ordering policies with the effects of inflation and trade credit for deteriorating items in a two-warehouse setup. Palanivel et al. (2016) developed and incorporating non-instantaneous deterioration in a two-warehouse inventory model with stock- Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 257 https://internationalpubls.com dependent demand, inflation and shortages. Jaggi et al. (2017) presented a two-warehouse inventory model for deteriorating items and incorporated permissible delay in payments with imperfect quality. Kumar and Chanda (2018) introduced a model for two-warehouse where demand is influenced by innovation, relevant to growing technology markets. Panda et al. (2019) proposed a two-warehouse inventory model incorporating partial backlogging, credit policy along with price- and stock- dependent demand. Sethy et al. (2020) included a two-warehouse production prototype for managing deteriorating inventory items within payment structures. Xu et al. (2021) formulated a model for items with infinite lifespan and warehouse mode selection with partial backlogging. Qiuet al. (2022) presented an optimization approach management in a dual-channel warehouse for multi-product inventory. Das et al. (2023) developed a dual-channel supply chain model with all-units discount and partial backordering under a two-warehouse setting. Sharma et al. (2024) formulated a green inventory model and included carbon emission with energy consumption in two ware house system. Sharma and Mandal (2024) explored as a sustainable two-warehouse inventory models with preservation technology investment. Deterioration: Considering deterioration in an inventory model is crucial for maintaining realistic stock levels, optimizing costs, and ensuring customer satisfaction. Items like food, pharmaceuticals, and electronics degrade over time, and failing to account for this can lead to significant waste and financial loss. By integrating deterioration rates, businesses can make informed decisions about order quantities and timing, reducing holding costs and minimizing spoilage. This approach also supports regulatory compliance and enhances supply chain efficiency, ensuring that high-quality products are available when needed, ultimately fostering customer loyalty and trust. Bhunia et al. (2015) introduced an inventory model of two storage for perishable items incorporating variable demand. Pervin et al. (2016) formulated an inventory model tailored for perishable items in markets with declining demand, emphasizing the role of trade credit policy. Chan et al. (2017) integrated production and inventory management for deteriorating items, focusing on optimizing production rates while accounting for deterioration during delivery. Singha et al. (2018) presented a fuzzy model for decaying items along with stock-dependent demand rates. This approach allows for better handling of uncertainties in inventory levels and demand fluctuations, crucial for managing perishable products effectively. Braglia et al. (2019) introduced a continuous review inventory model for decaying items under uncertain demand and lead time. Khakzad and Gholamin (2020) explored an inventory model for deteriorating items with the effect of inspection on the deterioration rate. Sharma et al. (2021) described a production-based model for deteriorating items with price discount and inventory dependent demand. Further, Ghandehari and Karimi-Lenji (2022) formulated an optimal inventory policy for perishable items with a multivariate demand, addressing complex demand patterns and their effect on inventory management. Kumar et al. (2022) developed an inventory model with advertisement and price-based demand function along with deterioration. Mahato et al. (2023) presented inventory models for decaying items with fixed lifetimes and carbon emissions policies, which highlight the integration of environmental considerations into inventory management for deteriorating goods. Recently, San-Jose et al. (2024) proposed a sustainable inventory model under a carbon emission tax for perishable items with full backlogging. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 258 https://internationalpubls.com Trade Credit Period: Incorporating trade credit into an inventory model is essential for optimizing financial management and improving supplier relationships. Trade credit allows businesses to delay payments for inventory purchases, providing crucial cash flow flexibility. This can enable companies to invest in other areas, such as marketing or expansion, while still maintaining adequate stock levels. By including trade credit terms in inventory models, businesses can more accurately assess the true cost of holding inventory, balance their working capital, and negotiate better terms with suppliers. This strategic financial tool helps in managing liquidity and can lead to more favorable purchasing conditions, enhancing overall operational efficiency. Sarkar et al. (2015) introduced an inventory model incorporating trade-credit period and variable deterioration rate. Mahata and De (2016) presented an EOQ inventory system for repairable items with price-dependent demand rates under a partial trade credit policy for retailer. This model offers insights into optimizing order quantities and credit utilization in inventory management. Tsao et al. (2017) explored explores sustainable newsvendor models under trade credit, emphasizing the environmental and financial benefits of integrating sustainability considerations into inventory decision-making. Tiwari et al. (2018) developed an inventory model with expiration dates for deteriorating items under two-level partial trade credits and with partial backlogging. It addresses the complexities of managing perishable goods efficiently. Pervin et al. (2019) proposed a two-echelon inventory model incorporating a trade-credit policy with price- and stock-dependent demand. Kumar et al. (2020) explored a model and included trade credit policy for deteriorating items and multivariable demand function. Barron et al. (2020) presented an inventory model with stock-dependent carrying costs and non-linear inventory-dependent demand with a trade credit period. This model accounts for the dynamic nature of inventory costs and demand fluctuations. Esmaeili and Nasrabadi (2021) proposed a model for multi-retailers consisting of trade credit and an inflationary environment. Sharma et al. (2022) explored an economic quantity model for decaying items under the effect of inflation along with trade credit policy. Shan and Shroff (2022) introduced a model for fixed-life products with a two-level trade credit policy and trapezoidal demand. Moradi et al. (2023) considered learning effects and partial trade credit policy in their study and presented an inventory model for imperfect quality items. Shah et al. (2024) proposed a model for non-instantaneous deteriorating items with advertisement-dependent probabilistic demand under trade credit financing. Inflation: Incorporating inflation into an inventory policy is vital for maintaining accurate cost assessments and ensuring profitability. Inflation affects the purchasing power of money, causing prices of goods to rise over time. By accounting for inflation, businesses can better forecast future costs, adjust pricing strategies, and optimize order quantities to mitigate the impact of rising prices. This proactive approach helps in preserving profit margins, maintaining competitiveness, and ensuring long-term financial stability. Pal et al. (2015) formulated a production model with inflation along with shortage and ramp type demand. A bi-objective inventory model under inflation and discount was introduced by Mousavi et al. (2016). Yadav et al. (2017) investigated the effect of inflation on a two- warehouse model with time dependent demand. Shah and Vaghela (2018) proposed a production model for effort and time-based demand under inflationary environment. Yadav et al. (2019) presented a supply chain model under inflationary conditions for deteriorating items. It offers insights into optimizing inventory across the supply chain network. Further, Kumar et al. (2020) and Sundararajan et al. (2021) analyzed the effect of inflation in their various inventory models. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 259 https://internationalpubls.com Recently Sarkar et al. (2022) applied neural network within an inventory model under uncertainty and inflation. It leverages advanced computational techniques to improve inventory decision-making accuracy. The effect of inflation with multivariate demand on the EOQ model along with partial backlogging and carbon tax policy was introduced by Singh et al. (2023). Kumar et al. (2023) investigated the combined effect of promotional efforts and selling price in an inventory model under inflation. Further Pal et al. (2024) proposed an inventory model of two-warehouse with credit policy and inflation effect. It addresses the financial aspects of inventory management in a multi- warehouse setting, considering inflationary pressures. In this study we have formulated an inventory model for decaying items with two warehouse one is rented and another is owned. Demand is dependent on price and advertisement. Shortage is allowed and partially backlogged. Holding cost is variable in nature and time dependent. Additionally, various costs are considered in inflationary environment. To increase the flow of money a trade credit period is also offered to the retailer. Using examples and careful analysis, the study showed how these choices affect the overall cost and offered helpful tips for managers. Overall, it helps improve how companies manage their supply chains for perishable goods, making operations more efficient and cost-effective. Table-1 Comparison table between Previous research work and current work. References Variable Demand Variable Holding cost Shortage Inflation Deterio ration Trade credit Jaggi et al. (2015) Yes No No Yes Yes No Bhunia et al. (2015) Yes No Yes No Yes No Tiwari et al. (2016) No No No Yes Yes Yes Palanivel et al. (2016) Yes No Yes Yes Yes No Jaggi et al. (2017) No No No No Yes Yes Kumar and Chanda (2018) Yes No No No Yes No Panda et al. (2019) Yes No Yes No Yes No Sethy et al. (2020) Yes No No No Yes Yes Xu et al. (2021) Yes No Yes No No No Ghandehariet al. (2022) Yes No No No Yes No Das et al. (2023) Yes No Yes No No Yes San-Jose et al. (2024) Yes No No No Yes No Pal et al. (2024) No No Yes Yes Yes Yes Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 260 https://internationalpubls.com This Paper Yes Yes Yes Yes Yes Yes Fig.1 Graphical Representation of presented model 3. Assumptions and list of symbols: The manuscript uses the following conventions and symbols to mathematically define the proposed inventory procedure. 3.2 Assumptions: • To maintain the inventory level model is based on two warehouses with fixed capacity, one is rented (RW) and another is owned warehouse (OW). • Demand is based on market advertisement and price of product that is 𝐷 = 𝐴𝛼(𝑎 − 𝑏𝑝), where 𝑎, 𝑏 and 𝛼 are demand parameter, A is the frequency of market advertisement and 𝑝 is the product selling price. • Shortages are allowed in the time period [𝑡2, 𝑇], which is partially backlogged in the proportion 𝑒−𝛿(𝑇−𝑡) of demand where 0 < 𝛿 < 1. • 𝜃1 is the deterioration rate of inventory in the rented warehouse while 𝜃2 is the deterioration rate of owned warehouse. • Holding cost of inventory is variable and time dependent. • Replenishment rate of inventory is infinite and lead time is zero. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 261 https://internationalpubls.com • In the model, deteriorating items are neither repaired nor replaced throughout the cycle length [0, T]. • Various costs are taken in the inflationary environment. • Trade credit period 𝑀 is offered to the supplier to improve the business. 3.2 List of symbols • 𝑎, 𝑏 Demand parameters 𝑎 > 0, 𝑏 > 0 (constant) • 𝐴 Advertisement frequency • ℎ0 + ℎ1𝑡 variable holding cost for rented warehouse • ℎ2 + ℎ3𝑡variable holding cost for owned warehouse • 𝐼𝑒 Rate of Interest earned • 𝐼𝑐 Rate of Interest charged • 𝑝- Market price ($/unit) • 𝑅Maximum backlogged amount (Per order) • 𝑆Inventory in the owned warehouse initially (units) • 𝑄 Total level of inventory in the both warehouses at time 𝑡 = 0(units) • 𝑄 − 𝑆 Initial stock level of rented warehouse(units) • 𝑇𝐶Total cost per cycle for the inventory procedure ($/time unit) • 𝛿 Backorder parameter. 𝛿 > 0 • 𝜃1Rate of deterioration of rented warehouse, 0 < 𝜃1 < 1(in %) • 𝜃2Rate of deterioration of owned warehouse, 0 < 𝜃2 < 1(in %) • 𝐼𝑟(t) Inventory in rented warehouse at any time t(units) • 𝐼𝑜1 (t) Inventory in owned warehouse in time interval [0,𝑡1] (units) • 𝐼𝑜2 (t) Inventory in owned warehouse in time interval [𝑡1,𝑡2,] (units) • 𝐼𝐵(t)backorder inventory during shortage in time interval [𝑡2,𝑇] (units) • 𝑀Offered trade credit period • 𝐶𝑙 Lost sale cost($/order) • 𝐶𝑑 Deterioration cost($/order) • 𝐶𝑠 Shortage cost($/order) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 262 https://internationalpubls.com 3.3 Decision variables • 𝑡1Time at which stock of rented warehouse become zero • 𝑡2Time at which total stock of owned warehouse become zero • 𝑇Cycle duration (time unit) 4. Mathematical Modeling: Let us consider in this model 𝑄 units of inventory received at time 𝑡 = 0. Out of which S units of inventory are stored in the owned warehouse 𝑂𝑊 and remaining 𝑄 − 𝑆 units are kept in rented warehouse 𝑅𝑊. Now to meet the demand inventory of rented warehouse will be consumed first. Let in the time period [0, 𝑡1] inventory of rented warehouse become zero due to deterioration and demand. And inventory of 𝑂𝑊 depletes due to deterioration only during this period [0, 𝑡1]. Fig.2 Two warehouse inventory model R Further during the period [𝑡1, 𝑡2] inventory of 𝑂𝑊 gets down due to demand and deterioration and become zero at 𝑡 = 𝑡2. After this shortage started during time period [𝑡2, 𝑇] which is partially backlogged and 𝑅 is the maximum shortage inventory. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 263 https://internationalpubls.com Following differential equations are the representation of inventory level at any time 𝑡 in 𝑅𝑊 𝑎𝑛𝑑 𝑂𝑊 in the duration [𝑡2, 𝑇] . 𝑑𝐼𝑟(𝑡) 𝑑𝑡 + 𝜃1𝐼𝑟(𝑡) = −𝐴𝛼(𝑎 − 𝑏𝑝), 0 ≤ 𝑡 ≤ 𝑡1 (1) 𝑑𝐼𝑜1 (𝑡) 𝑑𝑡 + 𝜃2𝐼𝑜1 (𝑡) = 0, 0 ≤ 𝑡 ≤ 𝑡1 (2) 𝑑𝐼𝑜2 (𝑡) 𝑑𝑡 + 𝜃2𝐼𝑜2 (𝑡) = −𝐴𝛼(𝑎 − 𝑏𝑝), 𝑡1 ≤ 𝑡 ≤ 𝑡2 (3) 𝑑𝐼𝐵(𝑡) 𝑑𝑡 = −𝐴𝛼(𝑎 − 𝑏𝑝)𝑒−𝛿(𝑇−𝑡),𝑡2 ≤ 𝑡 ≤ 𝑇 (4) With boundary conditions 𝐼𝑟(0) = 𝑄 − 𝑆, 𝐼𝑟(𝑡1) = 0 , 𝐼𝑜1 (0) = 𝑆 , 𝐼𝑜2 (𝑡2) = 0, 𝐼𝐵(𝑡2) = 0 𝑎𝑛𝑑 𝐼𝐵(𝑇) = −𝑅 On solving above equation (1), (2), (3) and (4) we get 𝐼𝑟(𝑡) = 𝑄−𝑆 𝑒𝜃1𝑡 − 𝐴𝛼(𝑎−𝑏𝑝) 𝜃1 + 𝐴𝛼(𝑎−𝑏𝑝) 𝜃1𝑒𝜃1𝑡 , 0 ≤ 𝑡 ≤ 𝑡1 (5) 𝐼𝑜1 (𝑡) = 𝑆𝑒−𝜃2𝑡, 0 ≤ 𝑡 ≤ 𝑡1 (6) 𝐼𝑜2 (𝑡) = 𝐴𝛼(𝑎−𝑏𝑝) 𝜃2 (𝑒𝜃2(𝑡2−𝑡) − 1), 𝑡1 ≤ 𝑡 ≤ 𝑡2 (7) 𝐼𝐵(𝑡) = 𝐴𝛼(𝑎−𝑏𝑝) 𝛿 (𝑒−𝛿(𝑇−𝑡2)) − 𝑒−𝛿(𝑇−𝑡)), 𝑡2 ≤ 𝑡 ≤ 𝑇 (8) Also 𝐼𝑟(𝑡1) = 0 therefore by (5) we get 𝑆 = 𝑄 − 𝐴𝛼(𝑎−𝑏𝑝) 𝜃1 (𝑒𝜃1𝑡1 − 1) (9) Applying the condition of continuity 𝐼𝑜1 (𝑡1) = 𝐼𝑜2 (𝑡1) we get 𝑆 = 𝐴𝛼(𝑎−𝑏𝑝) 𝜃2 (𝑒𝜃2𝑡2 − 𝑒𝜃2𝑡1) (10) putting this value in equation (9) we get 𝑄 = 𝐴𝛼(𝑎 − 𝑏𝑝) { 1 𝜃1 (𝑒 𝑒−𝑖𝑡1 𝑖 𝑡1 − 1) + 1 𝜃2 (𝑒𝜃2𝑡2 − 𝑒𝜃2𝑡1)} (11) In the next cycle the total quantity to be replenished is expressed as 𝑇𝑂𝑄 = 𝐼𝑟(0) + 𝐼𝑜(𝑡) − 𝐼𝐵(𝑡) 𝑇𝑂𝑄 = 𝐴𝛼(𝑎 − 𝑏𝑝) [ 1 𝜃2 (𝑒𝜃2𝑡2 − 𝑒𝜃2𝑡1 + 𝑒𝜃2(𝑡2−𝑡) − 1) − 1 𝛿 (𝑒−𝛿(𝑇−𝑡2)) − 𝑒−𝛿(𝑇−𝑡))] (12) Let us consider the ordering cost is 𝑂𝐶 Cost of holding inventory in Rented warehouse 𝐻𝐶𝑅 = ∫ (ℎ𝑜 + ℎ1𝑡)𝐼𝑟(𝑡)𝑒−𝑖𝑡𝑡1 0 𝑑𝑡 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 264 https://internationalpubls.com 𝐻𝐶𝑅 = 𝐴𝛼(𝑎−𝑏𝑝) 𝜃1 [(ℎ𝑜 + ℎ1𝑡1) ( 𝑒−𝑖𝑡1 𝑖 − 𝑒−𝑖𝑡1 𝜃1+𝑖 ) − ℎ1 ( 𝑒−𝑖𝑡1 (𝜃1+𝑖)2 − 𝑒−𝑖𝑡1 𝑖2 ) − ℎ𝑜 ( 1 𝑖 − 𝑒𝜃1𝑡1 (𝜃1+𝑖) ) + ℎ1 ( 𝑒𝜃1𝑡1 (𝜃1+𝑖)2 − 1 𝑖2)] (13) Holding cost in owned warehouse is given by 𝐻𝐶𝑜 = ∫ (ℎ2 + ℎ3𝑡)𝐼𝑜1 (𝑡)𝑒−𝑖𝑡𝑡1 0 𝑑𝑡 + ∫ (ℎ2 + ℎ3𝑡)𝐼𝑜2 (𝑡)𝑒−𝑖𝑡𝑡2 𝑡1 𝑑𝑡 = 𝐴𝛼(𝑎−𝑏𝑝) 𝜃2 [( ℎ2 𝜃2+𝑖 + ℎ3 (𝜃2+𝑖)2) (𝑒𝜃2𝑡2 − 𝑒𝜃2𝑡1 + 𝑒−𝑖𝑡1 − 𝑒−𝑖𝑡2) + ℎ3 (𝜃2+𝑖) (𝑡1𝑒−𝑖𝑡1 − 𝑡2𝑒−𝑖𝑡2) + ℎ2 𝑖 (𝑒−𝑖𝑡2 − 𝑒−𝑖𝑡1) + ℎ3𝑡2 𝑖 𝑒−𝑖𝑡2 + ℎ3 𝑖2 (𝑒−𝑖𝑡2 − 𝑒−𝑖𝑡1)] (14) Deterioration cost per cycle is 𝐷𝐶 = 𝐴𝛼(𝑎 − 𝑏𝑝)𝐶𝑑 [ 1 (𝜃1+𝑖) (𝑒𝜃1𝑡1 − 𝑒−𝑖𝑡1) + 1 (𝜃2+𝑖) (𝑒−𝑖𝑡1 − 𝑒𝜃2𝑡2 + 𝑒𝜃2𝑡1 − 𝑒−𝑖𝑡2) + 1 𝑖 (𝑒−𝑖𝑡2 − 1)] (15) Shortage Cost per cycle is 𝑆𝐶 = 𝐶𝑠 ∫ 𝐼𝐵(𝑡)𝑒−𝑖𝑡𝑇 𝑡2 𝑑𝑡 = 𝐶𝑠 𝐴𝛼(𝑎−𝑏𝑝) 𝛿 ( 𝑒−𝛿(𝑇−𝑡2)−𝑖𝑡2 𝑖 + 𝑒−𝛿(𝑇−𝑡2)−𝑖𝑡2 𝛿−𝑖 − 𝑒−𝛿(𝑇−𝑡2)−𝑖𝑇 𝑖 − 𝑒−𝑖𝑇 𝛿−𝑖 ) (16) Lost sale cost is given by 𝐿𝑆𝐶 = 𝐶𝑙 ∫ 𝐼𝐵(𝑡)𝑒−𝑖𝑡𝑇 𝑡2 𝑑𝑡 = 𝐶𝑙 𝐴𝛼(𝑎−𝑏𝑝) 𝛿 ( (𝑒−𝛿(𝑇−𝑡2)−𝑖𝑡2−𝑒−𝛿(𝑇−𝑡2)−𝑖𝑇) 𝑖 + (𝑒−𝛿(𝑇−𝑡2)−𝑖𝑡2−𝑒−𝑖𝑇) 𝛿−𝑖 ) (17) Advertising cost = 𝐴𝑜 Case – I When𝑴 ≤ 𝒕𝟐(permissible delay period is less than inventory period) In this condition, since positive stock period is larger than the credit period, the retailer can earn interest on the sales revenue at an annual rate 𝐼𝑒 in the time interval [0, 𝑡2]. ThereforeInterest earned is 𝐼𝐸1 = 𝑝𝐼𝑒 [∫ (𝑡1 − 𝑡)𝐴𝛼(𝑎 − 𝑏𝑝)𝑒−𝑖𝑡𝑑𝑡 + ∫ (𝑡2 − 𝑡)𝐴𝛼(𝑎 − 𝑏𝑝)𝑒−𝑖𝑡𝑑𝑡 𝑡2 𝑡1 𝑡1 0 ] = 𝑝𝐼𝑒𝐴𝛼(𝑎 − 𝑏𝑝) [ 𝑡1 𝑖 − 1 𝑖2 + 𝑒−𝑖𝑡2 𝑖2 + (𝑡2 − 𝑡1) 𝑒−𝑖𝑡1 𝑖 ] (18) And after this credit period 𝑀, retailer have to pay interest with annual interest rate 𝐼𝑟on the unsold stock and the payable interest 𝐼𝑃1 is Interest Payable 𝐼𝑃1 = 𝑝𝐼𝑝 ∫ 𝐼𝑜(𝑡) 𝑡2 𝑀 𝑒−𝑖𝑡𝑑𝑡 = 𝑝𝐼𝑝 𝐴𝛼(𝑎−𝑏𝑝) 𝜃2 [ 𝑒𝜃2(𝑡2−𝑀)−𝑖𝑀 𝜃2+𝑖 − 𝑒−𝑖𝑀 𝑖 − 𝑒−𝑖𝑡2 𝜃2+𝑖 + 𝑒−𝑖𝑡2 𝑖 ] (19) Therefore, total average cost is given by Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 265 https://internationalpubls.com 𝑇𝐶1(𝑡1, 𝑡2, 𝑇) = 1 𝑇 [𝑂𝐶 + 𝐻𝐶𝑅 + 𝐻𝐶𝑜 + 𝐷𝐶 + 𝑆𝐶 + 𝐿𝑆𝐶 + 𝐴𝐶 + 𝐼𝑃1 − 𝐼𝐸1] = 1 𝑇 [𝑂𝐶 + 𝐴𝛼(𝑎−𝑏𝑝) 𝜃1 [(ℎ𝑜 + ℎ1𝑡1) ( 𝑒−𝑖𝑡1 𝑖 − 𝑒−𝑖𝑡1 𝜃1+𝑖 ) − ℎ1 ( 𝑒−𝑖𝑡1 (𝜃1+𝑖)2 − 𝑒−𝑖𝑡1 𝑖2 ) − ℎ𝑜 ( 1 𝑖 − 𝑒𝜃1𝑡1 (𝜃1+𝑖) ) + ℎ1 ( 𝑒𝜃1𝑡1 (𝜃1+𝑖)2 − 1 𝑖2)] + 𝐴𝛼(𝑎−𝑏𝑝) 𝜃2 [( ℎ2 𝜃2+𝑖 + ℎ3 (𝜃2+𝑖)2) (𝑒𝜃2𝑡2 − 𝑒𝜃2𝑡1 + 𝑒−𝑖𝑡1 − 𝑒−𝑖𝑡2) + ℎ3 (𝜃2+𝑖) (𝑡1𝑒−𝑖𝑡1 − 𝑡2𝑒−𝑖𝑡2) + ℎ2 𝑖 (𝑒−𝑖𝑡2 − 𝑒−𝑖𝑡1) + ℎ3𝑡2 𝑖 𝑒−𝑖𝑡2 + ℎ3 𝑖2 (𝑒−𝑖𝑡2 − 𝑒−𝑖𝑡1)] + 𝐴𝛼(𝑎 − 𝑏𝑝)𝐶𝑑 [ 1 (𝜃1+𝑖) (𝑒𝜃1𝑡1 − 𝑒−𝑖𝑡1) + 1 (𝜃2+𝑖) (𝑒−𝑖𝑡1 − 𝑒𝜃2𝑡2 + 𝑒𝜃2𝑡1 − 𝑒−𝑖𝑡2) + 1 𝑖 (𝑒−𝑖𝑡2 − 1)] + 𝐶𝑠 𝐴𝛼(𝑎−𝑏𝑝) 𝛿 ( 𝑒−𝛿(𝑇−𝑡2)−𝑖𝑡2 𝑖 + 𝑒−𝛿(𝑇−𝑡2)−𝑖𝑡2 𝛿−𝑖 − 𝑒−𝛿(𝑇−𝑡2)−𝑖𝑇 𝑖 − 𝑒−𝑖𝑇 𝛿−𝑖 ) + 𝐶𝑙 𝐴𝛼(𝑎−𝑏𝑝) 𝛿 ( (𝑒−𝛿(𝑇−𝑡2)−𝑖𝑡2−𝑒−𝛿(𝑇−𝑡2)−𝑖𝑇) 𝑖 + (𝑒−𝛿(𝑇−𝑡2)−𝑖𝑡2−𝑒−𝑖𝑇) 𝛿−𝑖 ) + 𝐴𝑜 + 𝑝𝐼𝑝 𝐴𝛼(𝑎−𝑏𝑝) 𝜃2 [ 𝑒𝜃2(𝑡2−𝑀)−𝑖𝑀 𝜃2+𝑖 − 𝑒−𝑖𝑀 𝑖 − 𝑒−𝑖𝑡2 𝜃2+𝑖 + 𝑒−𝑖𝑡2 𝑖 ] − 𝑝𝐼𝑒𝐴𝛼(𝑎 − 𝑏𝑝) [ 𝑡1 𝑖 − 1 𝑖2 + 𝑒−𝑖𝑡2 𝑖2 + (𝑡2 − 𝑡1) 𝑒−𝑖𝑡1 𝑖 )]] (20) Case – II When 𝑴 > 𝒕𝟐 i.e permissible delay period is greater than inventory period As 𝑀 > 𝑡2, the retailer has to pay no interest and can accumulate interest at an annual rate 𝐼𝑒 during the time period (0, 𝑀)therefore earned interest 𝐼𝐸2 in this case is given by 𝐼𝐸2 = 𝑝𝐼𝑒 [∫ (𝑡1 − 𝑡)𝐴𝛼(𝑎 − 𝑏𝑝)𝑒−𝑖𝑡𝑑𝑡 + ∫ (𝑡2 − 𝑡)𝐴𝛼(𝑎 − 𝑏𝑝)𝑒−𝑖𝑡𝑑𝑡 𝑡2 𝑡1 𝑡1 0 + (𝑀 − 𝑡2) ∫ 𝑒−𝑖𝑡𝑑𝑡 𝑡2 0 ] = 𝑝𝐼𝑒𝐴𝛼(𝑎 − 𝑏𝑝) [ (𝑒−𝑖𝑡2−1) 𝑖2 + 𝑡1 𝑖 + (𝑡2−𝑡1) 𝑖 𝑒−𝑖𝑡1 − 1 𝑖2 − (𝑀−𝑡2)(𝑒−𝑖𝑡2−1) 𝑖 ] (21) And 𝐼𝑃2 = 0 Hence total average cost is given by 𝑇𝐶2(𝑡1, 𝑡2, 𝑇) = 1 𝑇 [𝑂𝐶 + 𝐻𝐶𝑅 + 𝐻𝐶𝑜 + 𝐷𝐶 + 𝑆𝐶 + 𝐿𝑆𝐶 + 𝐴𝐶 + 𝐼𝑃2 − 𝐼𝐸2] = 1 𝑇 [𝑂𝐶 + 𝐴𝛼(𝑎−𝑏𝑝) 𝜃1 [(ℎ𝑜 + ℎ1𝑡1) ( 𝑒−𝑖𝑡1 𝑖 − 𝑒−𝑖𝑡1 𝜃1+𝑖 ) − ℎ1 ( 𝑒−𝑖𝑡1 (𝜃1+𝑖)2 − 𝑒−𝑖𝑡1 𝑖2 ) − ℎ𝑜 ( 1 𝑖 − 𝑒𝜃1𝑡1 (𝜃1+𝑖) ) + ℎ1 ( 𝑒𝜃1𝑡1 (𝜃1+𝑖)2 − 1 𝑖2 )] + 𝐴𝛼(𝑎−𝑏𝑝) 𝜃2 [( ℎ2 𝜃2+𝑖 + ℎ3 (𝜃2+𝑖)2 ) (𝑒𝜃2𝑡2 − 𝑒𝜃2𝑡1 + 𝑒−𝑖𝑡1 − 𝑒−𝑖𝑡2) + ℎ3 (𝜃2+𝑖) (𝑡1𝑒−𝑖𝑡1 − 𝑡2𝑒−𝑖𝑡2) + ℎ2 𝑖 (𝑒−𝑖𝑡2 − 𝑒−𝑖𝑡1) + ℎ3𝑡2 𝑖 𝑒−𝑖𝑡2 + ℎ3 𝑖2 (𝑒−𝑖𝑡2 − 𝑒−𝑖𝑡1)] + 𝐴𝛼(𝑎 − 𝑏𝑝)𝐶𝑑 [ 1 (𝜃1+𝑖) (𝑒𝜃1𝑡1 − 𝑒−𝑖𝑡1) + 1 (𝜃2+𝑖) (𝑒−𝑖𝑡1 − 𝑒𝜃2𝑡2 + 𝑒𝜃2𝑡1 − 𝑒−𝑖𝑡2) + 1 𝑖 (𝑒−𝑖𝑡2 − 1)] + 𝐶𝑠 𝐴𝛼(𝑎−𝑏𝑝) 𝛿 ( 𝑒−𝛿(𝑇−𝑡2)−𝑖𝑡2 𝑖 + 𝑒−𝛿(𝑇−𝑡2)−𝑖𝑡2 𝛿−𝑖 − 𝑒−𝛿(𝑇−𝑡2)−𝑖𝑇 𝑖 − 𝑒−𝑖𝑇 𝛿−𝑖 ) + 𝐶𝑙 𝐴𝛼(𝑎−𝑏𝑝) 𝛿 ( (𝑒−𝛿(𝑇−𝑡2)−𝑖𝑡2−𝑒−𝛿(𝑇−𝑡2)−𝑖𝑇) 𝑖 + (𝑒−𝛿(𝑇−𝑡2)−𝑖𝑡2−𝑒−𝑖𝑇) 𝛿−𝑖 ) + 𝐴𝑜 + 0 + 𝑝𝐼𝑒𝐴𝛼(𝑎 − 𝑏𝑝) [ (𝑒−𝑖𝑡2−1) 𝑖2 + 𝑡1 𝑖 + (𝑡2−𝑡1) 𝑖 𝑒−𝑖𝑡1 − 1 𝑖2 − (𝑀−𝑡2)(𝑒−𝑖𝑡2−1) 𝑖 ]] (22) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 266 https://internationalpubls.com Hence total average cost is 𝑇𝐶(𝑡1, 𝑡2, 𝑇) = { 𝑇𝐶1(𝑡1, 𝑡2, 𝑇) 𝑀 ≤ 𝒕𝟐 𝑇𝐶2(𝑡1, 𝑡2, 𝑇), 𝑀 > 𝒕𝟐 5. Solution Procedure: By using equation (10) we can derive𝑇𝐶(𝑡1, 𝑡2, 𝑇) = 𝑇𝐶(𝑡2, 𝑇) Now the solution algorithm is expressed by the following flowchart. 6. Numerical Example: To authenticate the presented model, we employ the proposed algorithm to address the subsequent numerical instances. Mathematica 13.0 is utilized to ascertain the outcomes. Example-1 When 𝑴 ≤ 𝒕𝟐 The example is solved using the following inputs: Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 267 https://internationalpubls.com 𝐴 = 3, 𝛼 = 0.4. , 𝑎 = 110, 𝑝 = 50$ 𝑢𝑛𝑖𝑡 , 𝑏 = 2, 𝐴𝑜 = 50$ 𝑐𝑦𝑐𝑙𝑒 , 𝜃1 = 0.09 %, 𝜃2 = 0.13 %, ℎ𝑜 = 500$ 𝑐𝑦𝑐𝑙𝑒 , ℎ1 = 500 $ 𝑐𝑦𝑐𝑙𝑒 , ℎ2 = 40$ 𝑐𝑦𝑐𝑙𝑒 , ℎ3 = 40$ 𝑐𝑦𝑐𝑙𝑒 , 𝑖 = 0.8, 𝐶𝑠 = 5$ 𝑢𝑛𝑖𝑡 , 𝐶𝑙 = 25$ 𝑢𝑛𝑖𝑡 , 𝐶𝑑 = 2 $ 𝑢𝑛𝑖𝑡 , 𝑆 = 50 𝑢𝑛𝑖𝑡𝑠, 𝛿 = 0.9, 𝑀 = 80 𝑑𝑎𝑦𝑠, 𝐼𝑝 = 0.016, 𝐼𝑒 = 0.012 , 𝑂𝐶 = 100 $/𝑜𝑟𝑑𝑒𝑟 The optimum value of the decision variables are 𝑡1 = 53.2418 𝐷𝑎𝑦𝑠, 𝑡2 = 113.207 𝐷𝑎𝑦𝑠, 𝑇 = 149.8113 𝐷𝑎𝑦𝑠 and the minimum total inventory cost is 942.9614 $ 𝑖. 𝑒.approximately 942 $ per order respectively Example-2 When 𝑴 > 𝒕𝟐 The example is solved using the following inputs: 𝐴 = 3, 𝛼 = 0.4. , 𝑎 = 110, 𝑝 = 50$ 𝑢𝑛𝑖𝑡 , 𝑏 = 2, 𝐴𝑜 = 50$ 𝑐𝑦𝑐𝑙𝑒 , 𝜃1 = 0.09 %, 𝜃2 = 0.13 %, ℎ𝑜 = 500$ 𝑐𝑦𝑐𝑙𝑒 , ℎ1 = 500 $ 𝑐𝑦𝑐𝑙𝑒 , ℎ2 = 40$ 𝑐𝑦𝑐𝑙𝑒 , ℎ3 = 40$ 𝑐𝑦𝑐𝑙𝑒 , 𝑖 = 0.8, 𝐶𝑠 = 5$ 𝑢𝑛𝑖𝑡 , 𝐶𝑙 = 25$ 𝑢𝑛𝑖𝑡 , 𝐶𝑑 = 2 $ 𝑢𝑛𝑖𝑡 , 𝑆 = 50 𝑢𝑛𝑖𝑡𝑠, 𝛿 = 0.9, 𝑀 = 120 𝑑𝑎𝑦𝑠, 𝐼𝑝 = 0.016, 𝐼𝑒 = 0.012 , 𝑂𝐶 = 100 $/𝑜𝑟𝑑𝑒𝑟 The optimum value of the decision variables are 𝑡1 = 62.3412 𝐷𝑎𝑦𝑠, 𝑡2 = 109.237 𝐷𝑎𝑦𝑠, 𝑇 = 154.483 𝐷𝑎𝑦𝑠 and the minimum total inventory cost is 847.2341 $ 𝑖. 𝑒. approximately 847 $ per order respectively. 7. Sensitivity Analysis: Table-2 Paramete r % Change in parameter Change in Value 𝑡1 𝑡2 𝑇 TC 𝐴𝑜 -40% 30 51.4096 61.335 109.422 988.7416 -20 % 40 52.8652 87.921 128.536 968.5614 0 % 50 53.2418 113.207 149.813 942.9614 20% 60 55.7458 143.352 151.797 931.2587 40% 70 56.8246 149.021 154.814 922.5687 𝑝 -40% 30 54.9087 114.625 155.985 938.4175 -20 % 40 54.0123 114.112 152.354 940.6574 0 % 50 53.2418 113.207 149.813 942.9614 20% 60 52.9584 112.869 147.365 944.2457 40% 70 51.5647 112.013 145.214 946.5460 -40% 3 52.9876 110.568 149.1002 1548.5418 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 268 https://internationalpubls.com 𝐶𝑠 -20 % 4 53.0012 111.854 149.4657 1247.8714 0 % 5 53.2418 113.207 149.8113 942.9614 20% 6 53.4549 114.563 150.1236 658.4126 40% 7 53.6847 115.102 150.5487 342.1458 𝑀 -40% 48 53.5420 113.425 151.8385 754.6924 -20 % 64 53.3401 113.315 150.8364 845.9836 0 % 80 53.2418 113.207 149.8113 942.9614 20% 96 53.0913 113.109 148.7883 1037.9523 40% 112 52.9408 113.008 147.7635 1135.9253 𝛿 -40% 0.54 53.2497 113.389 150.5142 943.5544 -20 % 0.72 53.2455 113.298 150.1615 943.2551 0 % 0.90 53.2418 113.207 149.8113 942.9614 20% 1.08 53.2368 113.116 149.4740 942.6567 40% 1.26 53.2311 113.024 149.1235 942.3700 𝑂𝐶 -40% 60 53.1645 113.039 151.1459 1055.554 -20 % 80 53.2016 113.124 150.4786 998.2317 0 % 100 53.2418 113.207 149.8113 942.9614 20% 120 53.2928 113.295 149.1440 878.6900 40% 140 53.3514 113.375 148.4756 815.4201 𝐼𝑝 -40% 0.0096 53.1867 113.123 149.6488 1102.2471 -20 % 0.0128 53.2145 113.169 149.7748 1018.3574 0 % 0.0160 53.2418 113.207 149.8113 942.9614 20% 0.0192 53.2698 113.258 149.9369 870.5656 40% 0.0224 53.2988 113.305 1549.9832 795.1698 𝐼𝑒 -40% 0.0072 52.4743 112.540 149.3458 926.5847 -20 % 0.0096 52.8744 112.898 149.5748 935.7469 0 % 0.0120 53.2418 113.207 149.8113 942.9614 20% 0.0144 53.5982 113.624 149.9369 950.2315 40% 0.0168 54.024 114.036 150.1932 957.5024 Table-3 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 269 https://internationalpubls.com The Change of Directions in 𝑡1 , 𝑡2 , 𝑇 𝑎𝑛𝑑 𝑇𝐶 Due to change in Parameters. Change of Directions in Parameters Effect on 𝒕𝟏 , 𝒕𝟐 , 𝑻 𝒂𝒏𝒅 𝑻𝑪 𝒕𝟏 𝒕𝟐 𝑻 𝑻𝑪 𝐴0 ↑ ↑ ↑ ↑ ↓ 𝑝 ↑ ↓ ↓ ↓ ↑ 𝐶𝑠 ↑ ↑ ↑ ↑ ↓ 𝑀 ↑ ↓ ↓ ↓ ↑ 𝛿 ↑ ↓ ↓ ↓ ↓ 𝑂𝐶 ↑ ↑ ↑ ↓ ↓ 𝐼𝑝 ↑ ↑ ↑ ↑ ↓ 𝐼𝑒 ↑ ↑ ↑ ↑ ↑ To assess the impact of altering parameters within a range of -40% to +40%, a sensitivity table was constructed. This involved adjusting one variable at a time while maintaining all other parameters constant at their base values, as illustrated in the example. The resulting outcomes shed light on the ramifications of these adjustments on the total cost and time required to fulfil demand. • Increasing advertisement frequency results in higher total cost and time, suggesting a need for cost-benefit analysis in marketing strategies. • Lowering the selling price decreases total cost and time, and higher selling price increases the profit of business. • Increasing shortage costs result in higher total cost and time, indicating potential losses due to stockouts. • Shortening the trade credit period leads to a slight increase in total cost and time, i.e to decrease the total cost and make more profit trade credit period should be higher. • Changes in the backorder parameter have minimal effects on total cost and time, indicating the stability of the system under varying backorder conditions. • Decreasing ordering costs lead to lower total cost and time, emphasizing the importance of efficient procurement processes. • Reducing interest payable decreases total cost and time, indicating the significance of financial management in cost optimization. • Increasing interest earned leads to lower total cost and time, emphasizing the importance of maximizing returns on idle funds. Graphical representation is as follows: Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 270 https://internationalpubls.com 0 50 100 150 200 30 40 50 60 70 Ao Vs T T 850 900 950 1000 30 40 50 60 70 Ao Vs TC TC 135 140 145 150 155 160 30 40 50 60 70 p Vs T T 930 935 940 945 950 30 40 50 60 70 p Vs TC TC 148 148.5 149 149.5 150 150.5 151 3 4 5 6 7 Cs Vs T T 0 500 1000 1500 2000 3 4 5 6 7 Cs Vs TC TC 144 146 148 150 152 154 48 64 80 96 112 M Vs T T 0 500 1000 1500 48 64 80 96 112 M Vs TC TC Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 271 https://internationalpubls.com 8. Managerial Insights: Managers should carefully evaluate the impact of advertisement frequency and pricing strategies on total cost and time to optimize marketing expenditures. Efficient inventory management practices, including minimizing shortage costs and optimizing trade credit periods, can lead to cost savings and improved operational efficiency. Procurement processes should be streamlined to reduce ordering costs and enhance cost-effectiveness. Financial management strategies, such as 148 148.5 149 149.5 150 150.5 151 0.54 0.72 0.9 1.08 1.26 T 148 148.5 149 149.5 150 150.5 151 0.54 0.72 0.9 1.08 1.26 T 147 148 149 150 151 152 60 80 100 120 140 OC Vs T T 147 148 149 150 151 152 60 80 100 120 140 OC Vs T T 149.4 149.6 149.8 150 150.2 Ip Vs T T 0 500 1000 1500 Ip Vs TC TC 148.5 149 149.5 150 150.5 Ie Vs T T 910 920 930 940 950 960 Ie Vs TC TC Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 272 https://internationalpubls.com optimizing interest payable and maximizing interest earned, are crucial for minimizing total cost and improving profitability. 9. Conclusion: In this presented paper we have formulated an inventory model for instantaneous deteriorating items with two warehouse one is owned and another is rented. Holding cost is a variable and function of time 𝑡, which talks about how costs in different warehouses change over time and affect overall expenses. Holding cost of rented warehouse is higher than owned warehouse. Various costs are taken in the inflationary environment. Shortage is allowable and partially backlogged. By studying two different types of warehouses and considering partial backlogging (when orders can't be fulfilled completely) and inflation, the paper gives useful tips on how to make warehouses work better. Demand is taken multivariate function of advertisement and product price which talked about about how advertising, pricing, and promotions affect what people buy and how much, which is important for businesses. Trade credit period 𝑀is given to the retailer to enhance business. The goal of the research is to help businesses make smart decisions about things like how long to give credit, how much space to use in a warehouse, and how often to restock. It provides examples and analysis to help managers understand what choices might save them money and work best for their operations. By using Mathematica software to check its findings, the research makes sure its advice is trustworthy and can be used in real-world situations. Overall, it offers practical ideas to help businesses manage their warehouses better and save money. References [1] Bhunia, A. K., Shaikh, A. A., Sharma, G., &Pareek, S. (2015). A two storage inventory model for deteriorating items with variable demand and partial backlogging. Journal of Industrial and Production Engineering, 32(4), 263-272. https://doi.org/10.1080/21681015.2015.1046508 [2] Braglia, M., Castellano, D., Marrazzini, L., & Song, D. (2019). A continuous review,(Q, r) inventory model for a deteriorating item with random demand and positive lead time. Computers & Operations Research, 109, 102-121. https://doi.org/10.1016/j.cor.2019.04.019 [3] Cárdenas-Barrón, L. E., Shaikh, A. 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