Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 411 https://internationalpubls.com Optimizing Advertising and Pricing for Perishable Inventory with Freshness-Related Demand in a Hybrid Partial Prepayment and Trade Credit Supply Chain T.Vanjikkodi1*, V. Pankajam2 1Department of Mathematics, Sri G.V.G. Visalakshi College for Women, Udumalpet, India. vanjikkodi88@gmail.com 2Department of Mathematics, Sri G.V.G. Visalakshi College for Women, Udumalpet, India. pankajamgurusamy@gmail.com. Article History: Received: 16-05-2024 Revised: 23-06-2024 Accepted: 11-07-2024 Abstract: This paper investigates an inventory model addressing non-instantaneous deteriorating items with expiration concerns, incorporating a hybrid payment approach. The demand factors of this study are price-sensitive demand, product freshness considerations, and advertising frequency impact, which are critical elements in contemporary supply chain challenges. The proposed model investigates multiple prepayments and delayed payments to bolster business operations during financial emergencies. This study aims to enhance business flexibility, and the model reflects real-world complexities by introducing a time-dependent holding cost and accounting for partial backlogged shortages. The establishment of convexity ensures efficient optimization and numerical examples clearly illustrate how the proposed strategies influence overall enterprise profitability. Furthermore, the utilization of MATLAB for graphical representation enhances result availability and understanding. Sensitivity analysis enriches decision-making by providing important managerial insights during the recovery phase. This research contributes a strong foundation for inventory optimization that can adapt to changing post-pandemic economic conditions. Keywords: Price sensitive demand, Advertisement, Freshness of the goods, Trade Credit, Maximum lifetime, Hybrid payment. 1. Introduction In response to the worldwide pandemic, scholars have concentrated on creating effective inventory administration techniques that maximize revenue and reduce expenses, by considering the various payment possibilities. A customized inventory model that considers the market's state has been made, underlining how pricing significantly influences demand, especially when it comes to perishable commodities and expiry dates. The impact of advanced and deferred payment plans in the post pandemic period, on sales and overall inventory performance is also examined. Nowadays, consumers' awareness of food freshness and safety drives demand for products with expiration dates increased. They are underscoring the importance of effective advertising in creating awareness and meeting consumer needs. Well-crafted advertisements inform consumers about product attributes, evoke emotions, establish brand recognition, and positively influence purchasing decisions. This innovative inventory model integrates various dynamics, including non-instantaneous item deterioration, price-sensitive demand with advertising frequency and freshness of the products, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 412 https://internationalpubls.com and advanced payment strategies and discount rates, offering a comprehensive approach to inventory management that contributes significantly to the field's advancement. The initial motivating force is derived from understanding and experience with hybrid payment systems in real-world business situations. Advanced payment and payment discounts help to reduce consumer burdens, increase sales, and improve the economics of the supply chain. The second source of inspiration is the challenges of deteriorating goods. Prices fall, and storage costs rise when item quality deteriorates over time, adversely affecting the system's efficiency and financial advantages. The following motivating forces are the promotion of freshness of goods, advertisement for current products, emphasizing sensory appeal, health advantages, and quality control to appeal to customer preferences. Research through surveys and analyzing successful campaigns helps advertisers understand which freshness aspects connect best with different demographics, guiding effective messaging strategies to drive purchasing decisions. Another reason for this study is to address existing research gaps highlighted in the literature review. 2 Literature Review The literature review emphasizes the necessity for inventory models that accommodate the non- instantaneous deterioration of perishable items, advertisement frequency, product freshness, time- varying holding costs, trade credit, and advanced payment schemes. 2.1 Non-instantaneous Expiration date-related deterioration A rich and diverse body of research focuses on perishable inventory management, particularly considering the impact of expiration dates and deteriorating quality on supply chain dynamics. Managing a perishable inventory system with variable holding costs and non-instantaneous deterioration is essential in the retail, pharmaceutical, and agricultural sectors. Wu(2) defined this phenomenon as "Non-instantaneous deterioration (NID)". Hsu et al.([3]) introduced the idea of expiration within inventory models, highlighting the importance of managing perishable goods based on their maximum lifetime and deterioration rates. This phenomenon occurs frequently in everyday life, such as when some vegetables and fruits stay fresh for a short period with little or no rotting. Priyan and Mala(6) developed pricing models considering quantity discounts and demand elasticity influenced by perishable item deterioration and expiration dates, emphasizing the need for dynamic pricing strategies. Sebatjane and Adetunji(12) investigated supply chain models for perishable items where demand is influenced by expiration dates, highlighting the interconnectedness of supply chain participants in managing perishable inventories effectively. 2.2 Inventory model for Time varying holding cost Another notable aspect of inventory management is holding cost, which changes the rate of deterioration of products during storage. Still, holding costs may increase over time. The decision- maker may allocate additional resources to minimize the number of deteriorated items and decrease the associated losses incurred from these items. Researchers worldwide have developed several inventory models focusing on holding costs that vary over time. However, most researchers pursued their work in a typical environment rather than a hybrid partial prepayment and partial trade credit Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 413 https://internationalpubls.com scheme, which is crucial for retail businesses during the post-pandemic. As a result, in our suggested study, we attempt to overcome this gap. 2.3 Inventory model with an advanced payment In the post-pandemic situation, much research on hybrid payment schemes is needed. Changing company demands is necessary for improving inventory management systems since integrating advanced payment and trade credit mechanisms is critical for supply chain resilience. Prepaid service and (or) the discount percentage of the total purchase price for products by buyers to vendors under an inventory model with an advanced payment plan help stabilize markets in the face of erratic demand and supply. Scholars have studied prepayment systems with pioneering work by Zhang followed by innovative applications such as genetic algorithms for inventory models with advance payments and price discounts, as demonstrated by Gupta. Advanced payments on inventory management systems have influenced non-instantaneous deterioration and partial backlogging in recent years. Studies by Duary(15), Khan(16), and Tavassoli(18) have formulated inventory models integrating advanced payments with considerations like deterioration and partial back ordering alongside delayed payment policies. Additionally, The fixed lifetime with advanced payment and the discount rate was developed by Mukunda(20). 2.4 Inventory model with trade credit This research focuses on incorporating two payment schemes (Advanced payment and credit payment) with allowable shortages of items into an inventory model, representing an innovative approach within trade credit policies and inventory management. This approach adds complexity and realism to the model by considering different payment options and the possibility of item shortages, which can significantly impact supply chain dynamics and business operations. Furthermore, allowing shortages acknowledges the business's challenges in maintaining perfect inventory levels. It highlights the trade-offs between inventory costs, customer service levels, and financial constraints. Over the last twenty years, many researchers have incorporated trade credit in an inventory model. Some studies have suggested delaying payments partly to encourage retailers to place larger orders. Trade credit can stimulate the Inter-organization's power in the supply chain system developed by Lin(5). Mahato(9) established a trade credit policy-based sustainable EOQ model adopting partial back ordering shortages. Choudhury(14) investigated an integrated inventory model with capacity constraints under order-size-dependent trade credit, all-unit discount, and partial backlogging. In their study, they considered order size-dependent trade credit along with all unit discounts. Chung(21) created a model for managing inventory that considers products with varying deterioration rates over time and expiration dates while also considering that the retailer provides a credit period to its customers. Momena et al.(24) have recently devised a trade credit inventory model incorporating a quantity discount policy within a dual-storage facility. Previous studies predominantly explored inventory models incorporating trade credit payment schemes without simultaneously integrating multiple prepayment and trade credit policies alongside allowable shortages. In contrast, our research examines explicitly two payment schemes (advanced payment and credit payment) within an inventory model that realistically allows for item shortages, presenting a more comprehensive and practical approach to inventory management. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 414 https://internationalpubls.com 2.5 Price-dependent demand with advertisement This study focusing on an inventory model with a demand function incorporating a linearly decreasing price dependence and a nonlinear increase that is influenced by advertising and various additional factors such as prepayment, delay payment, partial backlogging, and time-varying deterioration with expiration date represents a comprehensive and timely approach to addressing real-world challenges in supply chain management; businesses are adapting to changing market conditions and consumer behaviours, especially after the pandemic. Market pricing is a critical aspect that is rigidly adhered to by market demand. Increasing the market price diminishes the market demand and vice versa. As a result, it plays a crucial role in inventory management. Connection with the Inventory model, where the selling price of the items governs the demand function, has been studied since then in the seminal work published by Whitin(1). Selling price-dependent demand with various conditions is a significantly trending research area for evidence in the last five years; Udayakumar et al.(13) developed delayed payment inventory models based on the assumption that non-instantaneous deterioration and demand is a deterministic function of selling price and advertising cost. This study highlights the complexities of perishable inventory management under extended supply chain dynamics. Many researchers, such as Khara et al.(8), San-Jose et al.(11), Alshanbari et al.(7), and Khan et al.(16) have investigated optimal advertising policies and hybrid payment mechanisms that combine advance and cash payments to maximize retailer profitability during business operations. Subhash(22) formulated a green product inventory model to address price-dependent demand, employing a teaching and learning-based optimization algorithm for solution refinement. Some of the related works are presented in Table 1. Table 1 Comparison of the study with related works Source Demand rate Deterioration Holding cost Advertisement Payment Scheme Discount on % Shortage Pre- Payment Delayed Payment [4] (2020) P & SD Constant Constant No Yes No Yes Constant & PB [7] (2021) PD TV & IS Constant Yes Yes No No PB [10] (2021) P & SD Constant Constant No Yes No Yes PB [17] (2022) P & TD Constant Constant No Yes Yes No No [16] (2022) TD TV & NIS LTD No Yes Yes No PB [19] (2023) PD with Freshness TV with expiration LTD Yes No No No PB [20] (2023) PD TV with expiration LTD with PC No Yes Yes Yes PB with WTD [23] (2024) PD Constant & NIS Constant Yes Yes No No PB This work PD with Freshness NIS & TVwith expiration LTD with PC Yes Yes Yes Yes PB with WTD Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 415 https://internationalpubls.com PD – Price Dependent, P & SD – Price and Stock Dependent, P & TD – Price and Time Dependent, TV – Time Varying, IS – Instantaneous, NIS – Non-Instantaneous, LTD – Linearly Time Dependent, PC – Purchase Cost, PB – Partial Backlogging, WTD – Waiting Time Dependent Based on the literature review, a research gap in inventory management was identified and addressed by formulating a comprehensive mathematical model. Price-sensitive demand, focusing on advertising frequency and product freshness, prepayment, delay payment, partial backlogging, and time-varying deterioration, considering non-instantaneous deterioration, are innovatively integrated into this model. This optimization model aims to maximize profit and underscores the significance of this research in the following research questions: 2.6 Research Questions 1. How do prepayment and delay-in payment schemes impact inventory management strategies and profitability? 2. How can we optimize balance inventory levels and partial backlogging to maximize profit while meeting demand? 3. What is the impact of considering time-varying deterioration with expiration dates to improve inventory decision-making for perishable goods? 4. How does price-sensitive demand, influenced by advertising frequency and product freshness, affect inventory policies and profitability? 5. What are this business strategy's practical challenges, implications, and market segments to ensure sustained profitability? This model fills a significant research gap and provides actionable outputs for businesses to optimize inventory practices, maximize profitability, and adapt to evolving market demands effectively. 3 Notations and assumptions The following notations and assumptions create a mathematical model of the proposed problem. 3.1 Notations The notations are as follows K - Replenishment cost per order c - Purchase cost per unit s - Shortage cost per unit d - Cost of deterioration per unit of time l - Lost sale cost per unit IE - Interest earned in a year L - Delivery lead-time M - Permissible delay in the payment period n - Number of installments defined for prepayments during the lead time Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 416 https://internationalpubls.com IL - Interest rate on loan per year IC - Interest rate charges per year td - Time at which the product remains fresh because of no deterioration α - Discount rate for purchasing cost due to prepayments S - Maximum stock per cycle R - Maximum shortage per cycle G - Expenditure per Advertisement Q - Order quantity per replenishment cycle i - Total profit for i = 1, 2, 3 Decision Variables t1 - Time at which the stock reaches zero p - Selling price per unit T - Cycle Time 3.2 Assumptions 1. There is an expiration date on items that are deteriorating. The degrading rate approaches the maximum lifetime ‘m’, at which point it approaches 1, which makes it practically relevant. To make this scenario possible, the rate of degradation is ( ) 1 ,0 1 t t T m m t  =    + − It may be noted that the product will not be sold after its maximum life of m. T, thus, needs to be less than or equal to m for the replenishing cycle time. 2. D(p) is the demand function that is contingent upon a tripartite relationship between advertising frequency, product freshness and selling price. Demand function expressed as ( ) ( ) m t D p A a bp m  −  = −     , where ‘A’ for the advertisement frequency,   [0, 1) is the elasticity of the advertising factor, ‘a’ for the scaling parameter and ‘b’ for the price sensitivity parameter and a − bp > 0. 3. Backorder has been used in this investigation. Given that some demand is backordered and this backorder rate is predicated on waiting time, it is naturally declining. The fraction is ( ) 1 ( ) 1 B T t T t − = + − , T− t representing the waiting time until the next replenishing point is reached and  is a backlogging parameter,  > 0. 4. The holding cost function can be expressed as: H(t) = g + ht, ‘g’ is the constant part of the holding cost, ‘h’ is the rate at which the holding cost increases per unit time of storage duration t. 5. The product is without deterioration between [0, td]; deterioration occurs at a variable rate θ(t) within the interval [td, t1]. 6. There is no lead-time and hence an endless replenishing rate. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 417 https://internationalpubls.com 7. Backlogged demand is immediately satisfied from inventory ordered in the previous cycle at the start of the subsequent cycle. 8. Damaged products are immediately replaced, repaired, or removed from stock. 9. During the lead period, the supplier returns the total purchase cost as a portion L and collects the remaining purchasing cost upon batch delivery. 10. The retailer secures a loan from a third party to cover prepayments, repaying it with principal and interest upon receiving consumer payment. 11. n equal instalments the retailers can pay a portion σcQ of the purchase price over L months before delivery, receiving a discount from the supplier. Shorter installment terms result in higher discount rates (α), with a maximum discount of J% for full upfront payment. 12. Planning horizon infinite. 4 Mathematical Formulation This research presents a mathematical model that integrates inventory management with payment strategies within a supply chain framework and focuses on hybrid prepayment and trade-credit policies, considering factors like shortages and gradual product deterioration. Specifically, the retailer pays a portion (σ) of the purchase price in n equal installments within a specified delivery lead time (L). The purchase price's remaining portion (1−σ) is settled upon order receipt (t = 0). Additional discounts or rebates (J %) to encourage the retailer's prepayment offers to the supplier, particularly in challenging circumstances such as during a pandemic. The supplier delivers goods to the retailer based on a predetermined payment schedule in the supply chain management. Also, the retailer determines replenishment cycles and pricing strategies by considering the payment terms and market conditions. Ultimately, clients receive retail products as part of this supply chain ecosystem. Figure 1. Graphical representation of the inventory system The model integrates an Economic Order Quantity (EOQ) framework to optimize order quantities and maximize the total profit of the inventory system. The inventory level I(t) is influenced by demand and deterioration, following a declining pattern over time. Specifically, the stock of this model decreases based on the demand rate from 0 to td. At t = t1, continuous demand depletes the inventory to zero. Subsequently, the supply experiences partial backlogging due to shortages between t1 and T. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 418 https://internationalpubls.com Differential equations of the inventory system can be described as ( ) ( )1 d, 0 t t dI t A a bp dt = − −   (1) with I1(0) = Imax. After solving (1), the inventory level I1(t) is obtained as ( ) ( )1 maxI t I A a bp t= − − (2) During the period [td, t1], the inventory level drops due to inventory-reliance demand and degradation. As a result, the following differential equation is used to represent the inventory system at any given time t ( ) ( ) ( )2 2 d 1 1 , t t t 1 dI t m t I t A a bp dt m t m  −  + = − −    + −   (3) with I2 (t1) = 0. The inventory level I2(t) is derived by solving (3). ( ) ( ) ( ) ( ) ( ) 1 2 1 1 1 1 1 A a bp m t I t m t t t m t In m m t  − + −   = + − − + + −   + −   (4) Considering the continuity of I(t) at t = td, it follows from (2) and (4) that I1(td) = I2(td) gives the maximum level of inventory for each cycle as ( ) ( ) 1 max 1 1 1 1 d d d m t m t S I A a bp t t t In m m t     + − + −  = = − + − +     + −      (5) The demand at time t is partially backlogged as a fraction ( ) 1 1 T t+ − during the period [t1, T]. As a result, the differential equation used to represent the inventory system at any given time t (t1 ≤ t ≤ T) is as follows: ( ) ( ) ( ) 3 1, t t 1 dI t A a bp T dt T t   − = −   + − (6) with I3(t1) = 0. Solving (6), yields the inventory level I3(t) as: ( ) ( ) ( )( ) ( )( )3 11 1 A a bp I t In T t In T t     −  = + − − + −  (7) The maximum amount of demand backlogged during each cycle can be computed as ( ) ( ) ( )( )3 11 A a bp R I T In T t    −  = − = + −  (8) As a result, the ordered quantity Q can be determined using (5) and (8) as follows: Q = S + R (9) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 419 https://internationalpubls.com The system consists of the following relevant costs: 1. Ordering costs are the fixed expenses associated with each retailer's purchase order to replenish inventory. Ordering cost plays a significant role in decision-making for supply chain management and inventory control strategies. The ordering cost can be expressed simply as: Ordering Cost (OC) = K 2. The total number of advertisements during each inventory cycle is A, and the cost per advertisement is G. Therefore, the total advertising cost during each inventory cycle would be: Total Advertising Cost (AC) = AG 3. The retailer's total inventory holding cost over the planning period (0, t1), which is divided into two parts (0, td) and (td, t1) with corresponding inventory levels I1(t) and I2(t), considering the inventory holding cost for each part separately and then combining them gives the total cost incurred by the retailer to maintain inventory throughout the specified period. ( ) ( ) ( ) ( ) 1 1 2 0 d d t t t HC c g ht I t dt g ht I t dt    = + + +       ( )( ) ( ) ( ) ( ) ( )( 2 3 max max 1 11 2 3 d d d d A a bpt t c gI t gA a bp hI hA a bp g m t t t m     −  = − − − − − + + −     ( ) ( ) ( ) 2 2 3 3 2 21 1 1 1 1 1 1 1 1 2 1 2 3 4 1 d d d d t t t t g m t m t m t m t In m t        − − + − − + − + − + − − + − +         + −         ( ) ( ) ( ) ( ) 22 2 3 3 4 4 1 1 1 1 1 1 1 1 1 2 3 4 6 d d d d mt t t t t t h m t m t h t t   +     − − − + + − + − + + −               ( ) ( ) ( ) 2 32 2 3 3 2 3 1 1 1 1 11 1 12 9 1 2 2 3 3 d d d d d m mt t t t m t t t m In m t  + +    − − + −  + + − + − + + −        + −         (10) 4. The cost of deterioration over the interval [td, t1] for the retailer's inventory, where deterioration affects the inventory's value due to spoilage, wear-and-tear, obsolescence, and similar factors, formulate the expression based on the quantity of inventory that has deteriorated during this period. ( ) ( ) ( ) ( )( ) ( ) ( )1 2 2 11 2 1 1 1 1 1 2 d t d d d d d dt t tdA a bpm t m t DC d I t A a bp dt t t t m t In m m m t      −−  − + −  = − − = − − + + − −       + −         (11) 5. Shortage cost is calculated during the time interval t1< t ≤ T, where customers experience delays or unavailability of goods. Various components related to the consequences of inadequate inventory levels constitutes the shortage cost. The expression for shortage cost is thus: ( )( ) ( ) ( ) ( )( ) 1 3 1 1 1 1 T t sA a bp SC s I t dt T t In T t     −   = − = − − + −    (12) 6. The last sale cost (LSC) is calculated as the integral of unmet demand D(p) over the time interval t1 < t ≤ T, multiplied by the unit opportunity cost ‘l’. This mathematical representation Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 420 https://internationalpubls.com assesses the economic implications of customers seeking alternatives due to inventory shortages. It guides inventory management to enhance sales performance and competitiveness. ( ) ( ) ( ) ( ) ( )( ) 1 1 1 1 1 1 1 1 1 T t LSC l A a bp dt lA a bp T t In T t T t          = − − = − − − + −   + −     (13) 7. The overall sales income for the entire cycle length, taking into account the item's selling price p and demand function D(p) during the period of favourable stock (t1), expresses the total sales revenue as: ( ) ( ) ( ) ( )2 2 1 1 0 0 1 2 d dt t d m t SR p A a bp dt p A a bp dt pR pA a bp t t t pR m m   −    = − + − + = − − − +         (14) 8. The cyclic capital costs associated with prepayments made by the retailer, where a percentage ‘σ’ of the purchase price is borrowed and paid in ‘n’ installments, and the remaining balance ‘1−σ’ is paid at the time of delivery, derive the expression for interest costs incurred during the prepayment period. ( )1 2 Lc I Q n L CP n  + = (15) 9. The discount rate ‘α’ offered by the supplier on prepayment of the purchase cost, which varies depending on the number of payments ‘n’ and the maximum discount rate ‘J’, use the following mathematical formula: c JQ DIS n  = (16) Analyzing the impact of payment terms ‘M’ across different inventory scenarios helps retailers make informed decisions to optimize revenue, manage costs, and enhance customer satisfaction. By understanding the financial implications of payment terms on business performance, retailers can implement effective strategies to drive growth and profitability. Case I: M  td  t1  T The monthly interest rate IC charged to the retailer for maintaining inventory during the settlement period [M, t1] derives the formula for ‘IC’ based on the interest rate applied to the value of inventory held over this period. Let's formulate the expression for IC1 as the yearly interest rate: ( ) ( ) 1 1 1 2 d d t t C M t IC cI I t dt I t dt    = +       ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 1 max 1 1 11 1 2 2 d d C d d A a bpt M t t cI I t M A a bp m t t t m t m      −   − − = − − − + + − − + −             ( ) ( ) 3 3 2 21 1 1 1 1 1 1 2 1 3 4 1 d d d t t m t m t m t In m t      − + −  + − + − − + − +         + −        (17) The total interest earned IE by the retailer based on customer demand and the interest rate, ‘IE’ Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 421 https://internationalpubls.com during the period [0, M], where the retailer earns interest on all items sold to customers, derives a formula that considers the sales revenue generated during this period. It is formulate the expression for IE as the total interest earned: ( ) ( ) ( ) ( )( )1 1 0 0 0 1 1 M t M E eIE pI D p dudt pI D p In T t dt    = − + + −       ( ) ( ) ( ) 2 3 2 1 1 1 (1 2 2 3 E d M M M pI A a bp t M In T t m       = − − − − + + −      (18) Therefore, the inventory system for the total profit per unit of time for this scenario is defined as: 1(p, t1, T) = (SR − OC − AC − DC − HC − SC − LSC − CP + IE1 + DIS − IC1) / T (19) Case II: td  M  t1  T The cost of interest charges for unsold stock over the time [M, t1] is illustrated as follows ( ) 1 12 2 t C M IC cI I t dt=  ( ) ( ) ( ) ( ) 2 2 3 3 1 1 1 1 11 1 2 3 CcI A a bp t M t M m t t M m t m   −    − − = + − − + − +           ( ) ( ) 2 2 1 1 1 1 1 1 2 1 4 1 m t m t m M In m M  + −   − + − − + − +    + −    (20) The interest earned for this scenario is ( ) ( ) ( ) ( )( )2 1 1 0 0 0 1 1 M t M E EIE IE pI D p dudt pI D p In T t dt    = = − + + −       ( ) ( ) ( ) 2 3 2 1 1 1 (1 2 2 3 E d M M M pI A a bp t M In T t m       = − − − − + + −      (21) Thus, the inventory system for the total profit per unit of time for this scenario is represented as: 2(p, t1, T) = (SR − OC − AC − DC − HC − SC − LSC − CP + IE2 + DIS − IC2) / T (22) Case III: td  t1 M  T After M, there is no positive stock remaining; consequently, the interest charged as: IC3 = 0 (23) The retailer accrues additional interest on inventory from M to T, and the interest received on sold products goes up to M. ( ) ( ) ( ) ( )( )2 1 3 1 1 1 1 2 E MIn T tt IE pI A a bp M t t     + − = − − + − +     (24) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 422 https://internationalpubls.com Hence, the inventory system for the total profit per unit of time for this scenario is given as: 3(p, t1, T) = (SR − OC − AC − DC − HC − SC − LSC − CP + IE3 + DIS − IC3) / T (25) The optimal total profit function for this inventory system is expressed as: 1 1 1 1 2 1 1 3 1 1 ( , , ),0 ( , , ) ( , , ),0 ( , , ),0 d d d p t T M t t T p t T p t T t M t T p t T t t M T        =           (26) 4.1 Solution procedure The main objective of the presented model in above section to maximize the total cost for the retailer; it is necessary to found out the value of p, t1, and T, which maximizes the cost. Directly obtaining the required optimal values of the total profit function is impossible because all the total profit functions are highly nonlinear. Initialize all the needed inputs for the total profit cost values Πi(p, t1, T) for i = 1, 2, 3 without decision variables, Using MATLAB Solver to obtain the optimization of the total profit function Πi(p, t1, T) for i = 1, 2, 3 with the constraints and to get optimal values of p, t1 and T. to obtain the optimal values of the decision variables p, t1 and T, also find optimal order quantity and optimal profit cost are found using obtained optimal values. To prove that the total profit function Πi(p, t1, T) for i = 1, 2, 3 is concave, we need to verify that the second derivatives of Πi(p, t1, T) with respect to p, t1, and T are all negative (or non-positive) within the above obtained optimal decision variables. It is demonstrated step-by-step for each Πi(p, t1, T) as: 1. Define the Total Profit Functions Πi(p, t1, T) for i = 1, 2, 3. 2. Compute the second partial derivatives of each profit function Πi(p, t1, T) with respect to p, t1, and T. 3. Verify Concavity: For concavity in p, ensure that 2 1 2 ( , , ) 0i p t T p     for all p, t1, and T. For concavity in t1, ensure that 2 1 2 1 ( , , ) 0i p t T t     for all p, t1, and T. For concavity in T, ensure that 2 1 2 ( , , ) 0i p t T T     for all p, t1, and T. 4. Ensure the mixed partial derivatives 2 2 2 1 1 1 1 1 ( , , ) ( , , ) ( , , ) , andi i ip t T p t T p t T t p T p T t             also satisfy conditions that indicate concavity. 5. If all the second derivatives satisfy the concavity conditions (i.e. non-positive) within the feasible parameter space, then the total profit function Πi(p, t1, T) for i = 1, 2, 3 is concave. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 423 https://internationalpubls.com 6. Verify the condition 2 2 2 2 2 1 1 1 1 2 2 2 1 1 2 2 2 2 2 1 1 1 1 2 2 1 1 2 1 ( , , ) ( , , ) ( , , ) ( , , ) ( , , ) ( , , ) ( , , ) ( , , ) ( , , 2 i i i i i i i i i p t T p t T p t T p t T p t T T t p t T p t T p t T p t T t p T T p t p t                 −                                 − −                     + 2 2 1 1 1 1 0 ) ( , , ) ( , , )i iT p t T p t T T t t p T p                                         which is the Hessian matrix analysis to examine the concavity properties of the total profit functions. 7. Analyzing the derivatives of the given profit functions, determine whether Πi(p, t1, T) is concave for i = 1, 2, 3 under the specified conditions. This verification is crucial in understanding the optimization landscape and ensuring the validity of profit maximization strategies. 5. Numerical examples and Sensitivity Analysis Example 1: The data provided in this example demonstrates numerically, the outcomes of the model developed. Consider the values a = 90, b = 1.5, A = 4, K = Rs.620 per order, c = Rs.15 per unit, s = Rs.11per unit, d = Rs. 10 per unit, l = Rs. 12 per unit, h = Rs.0.05 per unit per month, g = Rs. 0.3 per unit per month, M = 0.4 month, td = 0.7 month, m = 3 month, A = 4,  = 0.6, G = 50,  = 0.5,  = 0.4, J = 1, n = 3, L = 0.3, IC = 0.2, IE = 0.07, IL = 0.05. Optimal values for the above data are: 1 = Rs.2512.26, p = Rs.27.71, t1 = 1.4991, T = 1.7656, S = 141.7648, R = 27.8385, Q = 169.6034 in appropriate units. The Hessian values H1 = −5.7217 < 0, H2 = 10008.9540 > 0 and H3 = −2291002.4621 < 0 conform the concavity. Example 2: Other parameters are the same as in example 1 except M = 0.9. The optimal values for the data are: 2 = Rs.2588.10, p = Rs.27.51, t1 = 1.4938, T = 1.7520, S = 142.2509, R = 27.1894, Q = 169.4403 in appropriate units, and the Hessian values H1 = −5.8008 < 0, H2 = 10381.8905 > 0 and H3 = − 2446368.8423 < 0 conform the concavity. Example 3: Other parameters are the same as example 1 except M = 2.3 and c = 14. Taking the above data, determine the optimal values as 3 = Rs.2706.21, p = Rs.25.77, t1 = 2.2068, T = 2.4286, S = 198.9472, R = 24.8118, Q = 223.7590 in appropriate units, and the Hessian values are H1 = −5.1182 < 0, H2 = 5594.1131 > 0 and H3 = −52067.6911 < 0 conform the concavity. Example 3 clearly shows that when the credit facility increases, the total profit increases. The optimum values of this model are p*, t1*, T*, S*, R*, Q* and * are obtained from example 3. The study has examined the impact of changes made to well-known characteristics such as inventory cost, degradation, and demand. The results are tabulated, ranging from −50 % to +50% from the original values. Furthermore, changes are only considered for one parameter at a time, while the others remain unaltered. The results of this study are displayed in Table 2. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 424 https://internationalpubls.com Table 2 Sensitivity analysis for some input parameters for case I Parameter % Change of parameter p t1 T S R Q 1 c −50% 27.90 1.5771 1.8356 146.5663 26.8925 173.4587 2443.59 −25% 27.81 1.5374 1.8000 144.1161 27.3702 171.4863 2477.62 +25% 27.60 1.4619 1.7320 139.4923 28.2878 167.7801 2547.51 +50% 27.49 1.4256 1.6992 137.2425 28.7290 165.9715 2583.36 h −50% 27.70 1.5001 1.7663 141.8822 27.8177 169.6999 2512.67 −25% 27.71 1.4996 1.7660 141.8016 27.8287 169.6303 2512.47 +25% 27.71 1.4987 1.7652 141.7354 27.8385 169.5740 2512.06 +50% 27.71 1.4982 1.7649 141.6987 27.8582 169.5569 2511.85 g −50% 27.65 1.5055 1.7693 142.4989 27.6245 170.1234 2518.52 −25% 27.68 1.5023 1.7675 142.1317 27.7366 169.8683 2515.39 +25% 27.74 1.4959 1.7638 141.3982 27.9500 169.3481 2509.14 +50% 27.76 1.4927 1.7619 141.0755 28.0600 169.1355 2506.03 IC −50% 27.45 1.5536 1.8094 146.9123 27.0011 173.9134 2536.03 −25% 27.58 1.5260 1.7872 144.3124 27.4275 171.7399 2524.03 +25% 27.83 1.4730 1.7446 139.3212 28.2334 167.5547 2500.73 +50% 27.95 1.4474 1.7241 136.9159 28.6235 165.5394 2489.43 IE −50% 27.65 1.5168 1.7773 143.3278 27.2993 170.6270 2504.66 −25% 27.68 1.5080 1.7715 142.5501 27.5694 170.1195 2508.44 +25% 27.74 1.4903 1.7597 140.9865 28.0970 169.0835 2516.11 +50% 27.77 1.4814 1.7537 140.2006 28.3545 168.5551 2519.99  −50% 28.21 1.5097 1.7832 140.3352 28.0831 168.4183 2424.99 −25% 27.96 1.5044 1.7744 141.0533 27.9638 169.0170 2468.43 +25% 27.46 1.4938 1.7569 142.4698 27.7174 170.1872 2556.49 +50% 27.21 1.4885 1.7482 143.1682 27.5906 170.7587 2601.12 J −50% 28.18 1.5224 1.7919 141.3828 27.7234 169.1062 2417.05 −25% 27.95 1.5107 1.7787 141.5556 27.7778 169.3334 2464.44 +25% 27.47 1.4876 1.7526 141.9662 27.8970 169.8632 2560.51 +50% 27.24 1.4761 1.7396 142.1089 27.9447 170.0536 2609.18  −50% 28.06 1.4319 1.8065 135.3023 39.4136 174.7159 2539.03 −25% 27.86 1.4708 1.7821 139.0293 32.6104 171.6397 2523.38 +25% 27.59 1.5207 1.7537 143.8797 24.2944 168.1741 2503.95 +50% 27.50 1.5376 1.7447 145.5186 21.5606 167.0792 2497.50  −50% 26.70 1.9785 2.3134 118.6573 23.4431 142.1004 1517.28 −25% 27.28 1.7147 2.0145 129.3204 25.5847 154.9051 1957.84 +25% 28.06 1.3122 1.5478 155.5248 30.1814 185.7063 3208.50 +50% 28.36 1.1459 1.3530 170.3567 32.5679 202.9247 4081.60 Table 3 Sensitivity analysis for some input parameters for case II Parameter % Change of parameter p t1 T S R Q 2 c −50% 27.92 1.5344 1.7944 143.4068 27.0225 170.4293 2497.32 −25% 27.72 1.5136 1.7728 142.7837 27.1122 169.8959 2542.48 +25% 27.31 1.4747 1.7320 141.7005 27.2670 168.9675 2634.18 +50% 27.10 1.4563 1.7127 141.2215 27.3517 168.5732 2680.73 h −50% 27.51 1.4947 1.7528 142.3175 27.1795 169.4970 2588.51 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 425 https://internationalpubls.com −25% 27.51 1.4942 1.7524 142.2805 27.1894 169.4699 2588.31 +25% 27.52 1.4933 1.7517 142.1701 27.2009 169.3710 2587.89 +50% 27.52 1.4929 1.7513 142.1405 27.2009 169.3414 2587.68 g −50% 27.46 1.5001 1.7557 142.9364 26.9729 169.9093 2594.42 −25% 27.49 1.4970 1.7539 142.5753 27.071 169.6525 2591.26 +25% 27.54 1.4906 1.7502 141.8828 27.3030 169.1858 2584.95 +50% 27.57 1.4874 1.7483 141.5147 27.4063 168.9213 2581.81 IC −50% 27.47 1.5121 1.7692 143.7789 27.1137 170.8926 2590.24 −25% 27.49 1.5028 1.7605 143.0042 27.1566 170.1608 2589.16 +25% 27.53 1.4850 1.7438 141.5116 27.2322 168.7438 2587.06 +50% 27.56 1.4766 1.7359 140.7579 27.2565 168.0144 2586.05 IE −50% 27.33 1.5481 1.7950 147.0509 26.2105 173.2614 2564.33 −25% 27.42 1.5209 1.7736 144.6471 26.7172 171.3663 2576.10 +25% 27.60 1,4665 1.7303 139.8345 27.6672 167.5017 2600.34 +50% 27.69 1.4392 1.7083 137.4151 28.1111 165.5262 2612.82  −50% 28.08 1.4783 1.7470 138.6258 27.7330 166.3588 2511.33 −25% 27.80 1.4861 1.7496 140.4157 27.4670 167.8828 2549.87 +25% 27.23 1.5012 1.7544 144.0287 26.9231 170.9518 2627.14 +50% 26.94 1.5085 1.7567 145.8515 26.6552 172.5067 2666.62 J −50% 27.98 1.5157 1.7770 141.7860 27.0989 168.8848 2492.23 −25% 27.75 1.5047 1.7645 141.9998 27.1460 169.1458 2539.96 +25% 27.28 1.4829 1.7397 142.4442 27.2420 169.6863 2636.66 +50% 27.05 1.4721 1.7274 142.6313 27.2826 169.9139 2685.64  −50% 27.88 1.4283 1.7929 135.7971 38.6224 174.4195 2614.08 −25% 27.67 1.4663 1.7686 139.5176 31.9039 171.4215 2598.87 +25% 27.39 1.5146 1.7401 144.3173 23.7071 168.0244 2580.07 +50% 27.30 1.5310 1.7311 145.9278 21.0090 166.9368 2573.85  −50% 26.50 1.9610 2.2863 118.6259 22.9563 141.5822 1566.55 −25% 27.08 1.7061 1.9970 129.6199 25.0265 154.6464 2019.03 +25% 27.88 1.3082 1.5362 156.0246 29.4234 185.4480 3302.44 +50% 28.21 1.1421 1.3423 170.7133 31.6825 202.3958 4197.97 Table 4 Sensitivity analysis for some input parameters for case III Parameter % Change of parameter p t1 T S R Q 3 c −50% 26.07 2.1972 2.4194 19.6556 24.6364 221.2920 2578.26 −25% 26.11 2.2147 2.4415 197.4188 25.0904 222.5091 2624.73 +25% 25.31 2.2291 2.4510 202.9076 25.1560 228.0636 2715.45 +50% 25.05 2.2454 2.4685 205.3628 25.4746 230.8374 2762.30 h −50% 25.51 2.2322 2.4521 201.9140 24.7969 226.7109 2671.40 −25% 25.51 2.2314 2.4516 201.8686 24.8290 226.6976 2671.03 +25% 25.52 2.2298 2.4505 201.7191 24.8753 226.5944 2670.28 +50% 25.58 2.2158 2.4383 200.5693 25.0242 225.5935 2668.62 g −50% 25.45 2.2369 2.4537 202.5322 24.5073 227.0394 2678.36 −25% 25.49 2.2335 2.4522 202.1049 24.6827 226.7876 2674.49 +25% 25.55 2.2284 2.4506 201.4640 25.0140 226.4780 2666.92 +50% 25.58 2.2267 2.4505 201.1919 25.1630 226.3549 2663.22 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 426 https://internationalpubls.com IE −50% 25.54 2.2638 2.4993 203.5101 26.4385 229.9486 2598.58 −25% 25.80 2.2527 2.4852 201.3618 25.9225 227.2843 2635.77 +25% 25.59 2.2060 2.4226 199.9472 24.3866 224.3337 2703.33 +50% 26.08 2.1800 2.4008 195.6069 24.4818 220.0888 2734.36  −50% 26.43 2.1555 2.3800 192.1725 24.6144 216.7869 2618.74 −25% 26.03 2.1967 2.4190 196.8587 24.6760 221.5347 2646.13 +25% 25.64 2.2677 2.4993 203.1346 25.9483 229.0829 2694.03 +50% 25.23 2.2661 2.5012 205.4693 26.6335 232.1027 2713.95 J −50% 26.09 2.1952 2.4176 196.4251 24.6429 221.0680 2575.32 −25% 26.10 2.2138 2.4405 197.4261 25.0873 222.5134 2623.27 +25% 25.25 2.2453 2.4659 204.1820 25.0593 229.2413 2718.42 +50% 25.04 2.2467 2.4699 205.4956 25.4927 230.9883 2765.24  −50% 25.99 2.1856 2.5164 196.4251 24.6429 221.0680 2575.32 −25% 25.69 2.2034 2.4681 197.4261 25.0873 222.5134 2623.27 +25% 25.43 2.2408 2.4317 204.1820 25.0593 229.2413 2718.42 +50% 25.48 2.2628 2.4320 205.4956 25.4927 230.9883 2765.24 Case I Case II 1 1.2 1.4 1.6 1.8 2 1 1.5 2 2.5 1400 1600 1800 2000 2200 2400 2600 t1 Figure 2. Concavity of the total profit function with respect to t1 and T T T o ta l P ro fi t 26.5 27 27.5 28 28.5 1 1.5 2 2.5 2300 2350 2400 2450 2500 2550 p Figure 3. Concavity of the total profit function with respect to p and T T T o ta l P ro fi t 1 1.2 1.4 1.6 1.8 2 1 1.5 2 2.5 1400 1600 1800 2000 2200 2400 2600 t1 Figure 4. Concavity of the total profit function with respect to t1 and T T T o ta l P ro fi t 26.5 27 27.5 28 28.5 1 1.5 2 2.5 2350 2400 2450 2500 2550 2600 p Figure 5. Concavity of the total profit function with respect to p and T T T o ta l P ro fi t Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 427 https://internationalpubls.com Case III Observations of the table and graphs Analysis of the observations of the study based on the changes in parameters and their effects on various decision variables and performance metrics, including total profit, at the provided parameter changes and resulting values are presented below. Graphs of all three cases visualize the concavity of the total profit function with various decision parameters. The changes in each parameter, the decision variables and total profit for all three cases are described below: Table 2 highlights that decreasing purchase costs, interest rates, backlogging, increasing prepayments, and advertising elasticity enhance inventory management profitability-however, minimal impact parameters include holding costs, maximum discount, and interest earned. Strategic adjustments in these factors can significantly affect profitability, emphasizing the importance of informed decision-making in inventory management. Table 3 highlights the significance of examining and refining financial terms, holding costs, and purchasing costs to reduce total expenses and increase profitability in inventory management. Lower financing costs, improve operational efficiency and boost cash flow by carefully modifying purchase prices, interest rates, and payment periods. Furthermore, balancing inventory levels and backlog criteria allows for efficient demand fulfillment while reducing expenses, ultimately boosting supply chain operation's profitability and competitiveness. Table 4 outlines the strategic factors that decision-makers should take into account. It emphasizes the significance of financial management and cost optimization, as well as the awareness of how different parameters interact to affect profitability. It offers ways to increase profitability through financial planning, inventory control techniques, and successful negotiating tactics. Case III emphasizes the critical role of cost optimization and financial management in maximizing profitability. It demonstrates that managing purchasing costs and optimizing financial parameters significantly impact profits. Among the cases, Case III is likely to yield the highest profit by refining inventory control techniques and economic strategies. This efficiency in inventory management leads to improved financial arrangements and market competitiveness for the retailer. 2 2.2 2.4 2.6 2.8 2.2 2.4 2.6 2.8 3 2200 2300 2400 2500 2600 2700 t1 Figure 6. Concavity of the total profit function with respect to t1 and T T T o ta l P ro fi t 25 25.5 26 26.5 2.2 2.4 2.6 2.8 3 2510 2515 2520 2525 2530 2535 2540 p Figure 7. Concavity of the total profit function with respect to p and T T T o ta l P ro fi t Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 428 https://internationalpubls.com 6 Conclusion In this research article, the innovative attempt to provide a variety of payment options while operating an inventory which is subject to deterioration and sensitive demand has shown promising outcomes suitable for real-world situations. The effectiveness of inventory turnover in preserving product quality and boosting sales through dynamic controls and preservation techniques before deterioration is demonstrated numerically. This research has examined the effects of advanced payments and trade credit laws on a deteriorating product inventory model, accounting for factors that influence demand, such as product freshness, price, and advertisement frequency. The solution for the model clearly indicates the optimal selling price, cycle length, and shortage incidence period in order to maximize overall profit. The research also addresses the gaps appropriately, as mentioned above. It is evident that: Longer loan terms boost profitability, as the optimization process results in higher profits. The greatest revenue occurs when ( t1  M  T ). The significance of adopting a flexible approach to trade credit and degradation mitigation to maximize profit and expand the business through enhanced goodwill was underscored in this paper. The research outcomes can be briefed as follows: Enhanced Operational Reliability on trade credit and payback strategies provide timely stock delivery, improved cash flow, and stable finances. Sensitivity analysis, when combined with dynamic systems and precise demand forecasts, balances inventory levels, reduces the possibility of backlogs, and guarantees efficient operations. By optimizing shelf life management and accounting for time-varying degradation, choices about perishable inventory may be made more profitably and competitively in the market. Achieving maximum profitability in agile demand management for price-sensitive products may minimize waste through focused advertising and safety stock adjustments, boost customer satisfaction, and decrease waste and stockouts. Effective business strategies to meet the practical challenges are to sense the futuristic changes in demand and appropriately implement product pricing and payment options to ensure sustained profitability. The scope of the research encompasses the analysis of various real-world supply chain scenarios, specifically in the post-pandemic period, such as refurbishing items and handling products that are near the expiration date. Future research will explore various supply chain scenarios, including periodic price discount splits, product discounts, and stochastic demand. References [1] T. M. Whitin, Inventory control and price theory, Management Science, 1955;2: 61-68. [2] K.S. Wu, L.Y. Ouyang and C.T. Yang, An optimal replenishment policy for non- instantaneous deteriorating items with stock-dependent demand and partial backlogging, International Journal of Production Economics. 2006;101(2):369-384. https://doi.org/10.1016/j.ijpe.2005.01.010 [3] P.H. Hsu, H.M. Wee, and H.M. Teng, Optimal lot sizing for deteriorating items with expiration date, Journal of Information and Optimization Science. 2006;27(2): 271-286. https://doi.org/10.1080/02522667.2006.10699692 [4] M.A.A. Khan, A.A. Shaikh, G.C. Panda, I. Konstantaras and L.E. Cardenas-Barron, The effect of advance payment with discount facility on supply decisions of deteriorating products whose demand is both price and stock dependent, International Transactions in Operational Research,. http://refhub.elsevier.com/S1110-0168(23)00339-3/h0090 http://refhub.elsevier.com/S1110-0168(23)00339-3/h0090 http://refhub.elsevier.com/S1110-0168(23)00339-3/h0090 http://refhub.elsevier.com/S1110-0168(23)00339-3/h0090 http://dx.doi.org/10.1111/itor.12733 http://dx.doi.org/10.1111/itor.12733 http://dx.doi.org/10.1111/itor.12733 http://dx.doi.org/10.1111/itor.12733 http://dx.doi.org/10.1111/itor.12733 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 429 https://internationalpubls.com 2020;27:1343-1367. [5] B. Liu, Y. Wang and Y. Shou, Trade credit in emerging economies: An interorganizational power perspective, Industrial Management & Data Systems. 2020;120:768-783. [6] S. Priyan, and P. Mala, Optimal inventory system for pharmaceutical products incorporating quality degradation with expiration date: A game theory approach, Operations Research for Health Care. 2020;24:100245. https://doi.org/10.1016/j.orhc.2020.100245 [7] H.M. Alshanbari, A.A.A. H. El-Bagoury, M.A.A. Khan, S. Mondal, A.A. Shaikh and Rashid, Economic order quantity model with Weibull distributed deterioration under a mixed cash and prepayment scheme, Computational Intelligence and Neuroscience. 2021. https://doi.org/10.1155/2021/9588685. [8] B. Khara, J.K. Dey, and S.K. Mondal, An integrated imperfect production system with advertisement dependent demand using branch and bound technique, Flexible Services and Manufacturing Journal. 2021;33(2):508-546. [9] C. Mahato and G. C. Mahata, Optimal inventory policies for deteriorating items with expiration date and dynamic demand under two-level trade credit, Opsearch.2021;58:994-1017. [10] M.S. Rahman, M.A.A. Khan, M.A. Halim, T.A. Nofal, A.A. Shaikh and E.E. Mahmoud, 2021, Hybrid price and stock dependent inventory model for perishable goods with advance payment related discount facilities under preservation technology, Alexandria Engineering Journal. 2021;60(3):3455- 3465. https://doi.org/10.1016/j.aej.2021.01.045 [11] L.A. San-José, J. Sicilia, and B. Abdul-Jalbar, Optimal policy for an inventory system with demand dependent on price, time and frequency of advertisement, Computers & Operations Research. 2021; 128. https://doi.org/10.1016/j.cor.2020.105169 [12] M. Sebatjane, and O. Adetunji, Optimal lot-sizing and shipment decisions in a three-echelon supply chain for growing items with inventory level-and expiration date-dependent demand, Applied Mathematical Modelling. 2021;90:1204-1225. https://doi.org/10.1016/j.apm.2020.10.021 [13] R. Udayakumar, K.V. Geetha, and S.S. Sana, Economic ordering policy for non-instantaneous deteriorating items with price and advertisement dependent demand and permissible delay in payment under inflation, Mathematical Methods in the Applied Sciences. 2021;44(9):7697-7721. DOI:10.1002/mma.6594 [14] M. Choudhury, S.K. De and G.C. Mahata, Pollution-sensitive integrated production- inventory management for deteriorating items with quality loss and quantity loss with expiration date, International Journal of Systems Science: Operations & Logistics. 2022;9:546-568. https://doi.org/10.1080/23302674.2021.1950863 [15] Duary, S. Das, M.G. Arif, K.M. Abualnaja, M.A.A. Khan, M. Zakarya and A. A. Shaikh, Advance and delay in payments with the price-discount inventory model for deteriorating items under capacity constraint and partially backlogged shortages, Alexandria Engineering Journal. 2022;61(2):1735-1745. https://doi.org/10.1016/j.aej.2021.06.070 [16] M. A.-A. Khan, M. A. Halim, A. AlArjani, A.A. Shaikh and M.S. Uddin, Inventory management with hybrid cash-advance payment for time-dependent demand, time-varying holding cost and non- instantaneous deterioration under backordering and non-terminating situations, Alexandria Engineering Journal. 2022;61(11):8469-8486. https://doi.org/10.1016/j.aej.2022.02.006 [17] N. Manavizadeh, M. Shaabani and M. Rabani (2022), Jointly control of inventory and its pricing for a deteriorating item under multiple advance payments and delay in payments with partial backordering, Journal of Industrial and Systems Engineering. 2022 ;14:1-32. [18] S. Tavassoli, N. Manavizadeh, A. Rezaei and M. Rabbani, A lot-sizing model for non- instantaneous deteriorating products under advance payment and non-linear partial backlogging, Iranian Journal of Management Studies. 2022;15(1):85-110. [19] M.A.A. Khan, A.A. Shaikh, A.R. Khan and A.F. Alrasheedi, Advertising and pricing strategies of an inventory model with product freshness-related demand expiration date- related deterioration, Alexandria http://refhub.elsevier.com/S1110-0168(23)00339-3/h0175 http://refhub.elsevier.com/S1110-0168(23)00339-3/h0175 http://refhub.elsevier.com/S1110-0168(23)00339-3/h0175 http://refhub.elsevier.com/S1110-0168(23)00339-3/h0175 Health%20Care.%202020;24:100245 Health%20Care.%202020;24:100245 https://doi.org/10.1155/2021/9588685 http://refhub.elsevier.com/S1110-0168(23)00339-3/h0125 http://refhub.elsevier.com/S1110-0168(23)00339-3/h0125 http://refhub.elsevier.com/S1110-0168(23)00339-3/h0125 http://refhub.elsevier.com/S1110-0168(23)00339-3/h0125 http://refhub.elsevier.com/S1110-0168(23)00339-3/h0125 http://refhub.elsevier.com/S1110-0168(23)00339-3/h0125 http://dx.doi.org/10.1007/s12597-021-00507-7 http://dx.doi.org/10.1007/s12597-021-00507-7 http://dx.doi.org/10.1007/s12597-021-00507-7 https://doi.org/10.1016/j.aej.2021.01.045 http://refhub.elsevier.com/S1110-0168(23)00339-3/h0200 http://refhub.elsevier.com/S1110-0168(23)00339-3/h0200 http://refhub.elsevier.com/S1110-0168(23)00339-3/h0200 http://refhub.elsevier.com/S1110-0168(23)00339-3/h0200 http://refhub.elsevier.com/S1110-0168(23)00339-3/h0200 http://refhub.elsevier.com/S1110-0168(23)00339-3/h0215 http://refhub.elsevier.com/S1110-0168(23)00339-3/h0215 http://refhub.elsevier.com/S1110-0168(23)00339-3/h0215 http://refhub.elsevier.com/S1110-0168(23)00339-3/h0215 http://refhub.elsevier.com/S1110-0168(23)00339-3/h0215 http://refhub.elsevier.com/S1110-0168(23)00339-3/h0215 http://refhub.elsevier.com/S1110-0168(23)00339-3/h0260 http://refhub.elsevier.com/S1110-0168(23)00339-3/h0260 http://refhub.elsevier.com/S1110-0168(23)00339-3/h0260 http://refhub.elsevier.com/S1110-0168(23)00339-3/h0260 http://refhub.elsevier.com/S1110-0168(23)00339-3/h0260 http://refhub.elsevier.com/S1110-0168(23)00339-3/h0260 http://refhub.elsevier.com/S1110-0168(23)00339-3/h0260 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 5s (2024) 430 https://internationalpubls.com Engineering Journal. 2023;73:353-375. doi.rg/10.1016/j.aej.2023.04.059. [20] M. Choudhury and G.C. Mahata, Non-Instantaneous Deteriorating items inventory models with fixed Lifetime products under Hybrid Partial prepayment and Trade Credit in Supply chain, Journal of Industrial and Management Optimization. 2023;20(1):221-259. doi:10.3934/jimo.2023075 [21] K.J. Chung, S.F. Lee, H.M. Srivastava, S.D. Lin, S.L. Kang and J.J. Liao, Optimal ordering policy and preservation technology for deteriorating items with maximum lifetime under a resilient hybrid payment decision, Journal of Industrial and Management Optimization. 2023;19(7):5353-5379. Doi: 10.3934/jimo.2022176. [22] S.C. Das, H. Ali, M.A.A. Khan, A.A Shaikh and A.F. Alrasheedi, Inventory model for green products with payment strategy, selling price and green level dependent demand using teaching learning based optimization algorithm, Scientific Reports. 2024;14:3033. [23] S. Mandal, A. Banu and S.K. Mondal, An Integrated Inventory Model for non-instantaneous Deteriorating item Stochastic demand, RAIRO Operations Research. 2024;58(1):151-183. https://doi.org/10.1051/ro/2023087 [24] A.F. Momena, R. Haque, M. Rahaman and S.P. Mondal, A two storage inventory model with Trade credit policy and Time varying holding cost under quantity discount, MDPI logistics. 2024;7(4):77. https://doi.org/10.3390/logistics7040077 http://dx.doi.org/10.3934/jimo.2022176 http://dx.doi.org/10.3934/jimo.2022176 http://dx.doi.org/10.3934/jimo.2022176 http://dx.doi.org/10.3934/jimo.2022176 http://dx.doi.org/10.3934/jimo.2022176 https://doi.org/10.3934/jimo.2022176 https://doi.org/10.3390/logistics7040077