Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 1010 https://internationalpubls.com A Sustainable Two Warehouse Capacities based Inventory Model with Exponential Demand Pattern for Deteriorating Items Under Learning Effect Anchal1, Pushpendra Kumar1*, Alka Sharma3, A. K. Malik4 1Department of Mathematics, Shri Khushal Das University, Hanumangarh (Rajasthan) 2Department of Mathematics, Poornima University Jaipur, Rajasthan, INDIA 3School of Sciences, UP Rajarshi Tandon Open University, Prayagraj (U.P.) Email: anchal@skduniversity.com; pushpendra.kumar@skduniversity.com; alka.sharma@poornima.edu.in; profakmalik@uprtou.ac.in *Corresponding author Article History: Received: 04-03-2025 Revised: 25-04-2025 Accepted: 02-05-2025 Abstract Present paper deals a sustainable two warehouse based inventory mathematical model with exponential type of demand pattern for deteriorating items under effect of the learning where the deterioration rate follows the linear pattern and some carbon units emit from the electricity for the preserving of the deteriorating items. Two warehouse inventory models for degrading products are covered in this study. Physical products deterioration is a crucial component of every manufacturing and inventory system. Its influence can be disregarded if the pace of deterioration is relatively modest. On the other hand, degradation is a significant factor in many real-world scenarios. For the contemporary organisation, it is critical to manage and preserve the inventories of deteriorating assets. When degradation is time-dependent, the product's value decreases proportionately with time. Modellers were inspired to take the degradation factor into account as one of the modelling features upon understanding this factor. Deterioration rate is considered time dependent in this work. We take exponential demand rate into consideration in order to examine actual business scenarios and the learning effect involves in the holding cost, deterioration cost and ordering cost. The carbon emission cost also included in this study. We presented the numerical example for the justification of the model in this study, the cost reduction approach is applied. Keywords: Inventory, Two warehouses, exponential demand, learning effect. 1. INTRODUCTION The assumption of a constant demand rate, which served as the foundation for the 1915 development of the conventional economic order quantity (EOQ) model, was widely seen as being highly restricted and impractical. Actually, the continual demand assumption is only true up until the point of maturity. Inventory modellers gradually began to grasp the idea of time- varying needs. A uniform variation in the demand rate per unit of time is implied by a linear time changing demand rate. The EOQ model was only partially expanded by researchers to time-varying demand patterns. Donaldson (1977) developed a computer approach for mailto:anchal@skduniversity.com mailto:pushpendra.kumar@skduniversity.com mailto:alka.sharma@poornima.edu.in mailto:profakmalik@uprtou.ac.in Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 1011 https://internationalpubls.com determining the best reorder times with a linear demand trend over a finite planning horizon using elementary mathematics. Using the Silver–Meal heuristic (1969), Silver (1979) created an approximate solution approach for a linearly time-varying demand. A deterministic lot-size inventory model with shortages and a linear demand trend was put out by Dave in 1989. Goswami and Chaudhuri (1991) talked about a variety of inventory models that included a demand curve. Hill (1995) initially addressed the lack of discipline in the time-dependent demand pattern by seeing each order cycle as the result of the fusion of two distinct time periods, during which the demand is disciplined. Hariga (1995) examined how an inventory model with a time-dependent demand rate and shortages was affected by inflation and the time value of money. Deterministic models of perishable inventory with stock-dependent demand rate and nonlinear holding cost were presented by Bhunia A. K. and Maiti M. in 1998. A damageable item inventory with a variable replenishment rate and deterministic demand was examined by Mandal and Maiti (1997). One of the most pressing problems in the manufacturing and sale of goods is the control of inventory. Many companies fail each year due to the lack of adequate control of the inventories. Whether it is raw materials used to manufacture a product or products waiting to be sold, problems arise when too few or too many items are held in inventory. Inventory control techniques are very helpful in the development of inventory policies and in activity down the investment cost of inventories. Inventory control techniques not only help in minimizing the capital tied up in inventory but also provide the required service level. Study of deteriorating inventory model began with Ghare and Schrader (1963) who established the classical no shortage inventory model for constant rate of decay. Covert and Philip (1973) extended the above inventory model with a two-parameter Weibull distribution. An early discussion on an inventory model with two storage facilities is given by Hartely (1976). It is generally assumed that the holding cost in the RW is greater than the same in the OW. Hence, the items are stored first in the OW, and only excess of stock is stored in the RW. Further, the items of the RW are released first, and then the items of the OW. Dave and Patel (1981) considered an inventory model for deteriorating items with time proportional demand. Sarma (1983) developed a deterministic inventory model with finite replenishment rate. Sachan (1984) extended the model of Dave and Patel (1981) by allowing shortages. Dave (1988) further discussed the cases of bulk release pattern for both finite and infinite replenishments. He rectified the errors in Murdeshwar and Sathe (1985) and gives a complete solution for the model given by Sarma (1983). In the above literature, deterioration phenomenon was not taken into account. Assuming the deterioration in both warehouses, Sarma (1987), extended his earlier model to the case of infinite replenishment rate with shortages. A declining inventory model with stock-dependent demand and partial backlog was created by Dye (2002). An order level decaying inventory model with such time dependent quadratic demand was examined by Khanra and Chaudhuri (2003). An inventory model for degrading products with stock-dependent and time-varying demand rates over a constrained Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 1012 https://internationalpubls.com planning horizon was created by Balkhi & Benkherouf (2004). Stochastic lead times were taken into account in a manufacturing/remanufacturing system with predictable requests and returns by Zhou Y. W. and Yang S. (2005). Hou (2006) examined an inflation-adjusted inventory model for degrading goods with stock-dependent demand. There was a full backorder for the shortage. But in many real-world scenarios, the longer the waiting period is during a shortage period, the lower the backlog rate would be. Fuzzy-based optimization methods combine fuzzy logic with optimization processes. This helps handle uncertainty, vague information, and imprecise data in decision-making. These techniques use fuzzy sets and fuzzy numbers to represent uncertain data, which makes decision-making more flexible and effective in uncertain situations Kumar et al. (2023), Malik et al. (2012), Singh et al. (2014), Tyagi et al. (2023), Verma et al. (2022), Yadav and Malik (2014), Yadav et al. (2022). An EOQ model with fluctuating demand and deteriorating Weibull distribution was examined by Panda et al. in 2007. An inventory problem for non-instantaneously degrading commodities with stock-dependent demand was described by Ouyang L. Y. (2008) when the provider offered an all-unit quantity discount. Jayaswal et al. (2022) described a learning based inventory model for imperfect quality based inventory model under fuzzy environment. Alsaedi et al. (2023) presented carbon emissions based inventory model with fuzzy environment under learning effect. Jayaswal et al. (2021) generalised an inventory model with the effect of fuzzy under human learning. Deterministic developed an inventory model in this study with two warehousing facilities, degrading products under shortages, and deterministic and exponential demand. The two distinct warehousesβ€”one that is owned and the other that is rentedβ€”are thought to contain a single degrading object. The surplus units beyond the fixed capacity of W units of the OW are stored in the rented warehouse. Deterioration rates for both owned and rented warehouses are considered to be discrete, variable functions of time. 2. NOTATION AND ASSUMPTIONS 2.1. Assumptions The following assumption forms the basis of the single period inventory problem's mathematical model: 1. There is a set, restricted capacity of W units in the owned warehouse. 2. There is no limit to the size of the rented warehouse. 3. When RW is empty, OW things begin to be consumed. 4. Demand is time-dependent and predictable, involving D units per unit of time. evenly during the cycle duration T. 5. The expenses associated with inventory in RW, including storage and degradation costs, are greater than those seen in OW Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 1013 https://internationalpubls.com 6. There is no limit to the pace of replenishment, and it happens instantly. 7. If there are any backorders for excess demand at OW, shortages are allowed. 8. There is no lead time (zero). 9. In this scenario, the taxation policy is allowed. 10. The exchange policies of perishable items are not permitted. 11. The repair of the items is not allowed. 12. It is considered that carbon emissions come from the burning of the fuel. 13 It is also assumed, the deteriorating cost, holding cost (OW and RW) and ordering cost follow the effect of learning. 14. Shortages are allowed. 15. Preservation technology is also considered. 2.2 Notations In this paper, the notation used is as follows: 𝐷 = 𝐴𝑒𝐡𝑑Any given time t, the demand rate, where A and B are positive constants. 𝛼 + π‘Žπ‘‘, Inventory item degradation rate in OW where 0 < 𝛼 < 1, π‘Ž > 0 𝛽 + 𝑏𝑑,Rate of decline of inventory item in RW where 𝑏 > 0 π‘Žπ‘›π‘‘ 0 < 𝛽 < 1. π‘Š, The OW's maximum fixed storage capacity is W < S. 𝑇, whole preparation period. 𝑆𝐿1, {i: Cycle i} uses both OW and RW. 𝑆𝐿2, {j: In cycle j}, just OW is utilised. πΆπ‘œπ‘€ , carrying cost in OW for each inventory unit. πΆπ‘…π‘Š , carrying cost in RW for each inventory unit. 𝑆, The highest possible stock level at OW and RW. πΆπ‘œ, The cost of replenishment for each order. 𝐢𝑑, decline Price per unit. πΌπ‘œ(𝑑), At time t, the inventory level in OW. πΌπ‘Ÿ(𝑑), The RW value of the inventory at time t. 𝐢𝑠, Cost of shortage per unit of time. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 1014 https://internationalpubls.com 𝐾 = πΎπ‘œ + 𝐾1 𝑛𝑠, Ordering cost for the retailer; πΎπ‘œ, 𝐾1 = fixed and variable ordering cost; 𝑠 = Learning factor; 𝑛 = Number of shipment 𝐷𝑑 = 𝐷1 + 𝐷2 𝑛𝑠 ,The cost per unit item for the deterioration in OW/RW; π·π‘œ,𝐷2, Fixed and variable ordering cost; π»π‘Ÿ = 𝐻1 + 𝐻2 𝑛𝑠,The cost per unit item for the storage of items in RW; 𝐻1,𝐻2, Fixed and variable storage cost in RW; π»π‘œ = 𝐻3 + 𝐻4 𝑛𝑠, The cost per unit item for the storage of items in OW; 𝐻3, 𝐻4, Fixed and variable storage cost in OW; 𝑐1 Average vehicle fuel consumption when it is empty 𝐹𝑒 Carbon emissions due to electricity 𝑇π‘₯ Carbon taxation 3. MODEL DISCRIPTION We talk about the Deterministic Inventory Model for SL2 (system), at time t = 0, a lot size of S units, decaying with two warehouses where shortages happen at the conclusion of cycle. SL2, where W and S-W units are stored in OW and RW, respectively. The products of OW are only used up when RW runs out. The inventory S-W in RW declines as a result of degradation and demand throughout the time interval [o, t], and it disappears at t = t1 in OW. Only degradation causes the inventory W to drop, but only during [t1, t2]. Deterioration and demand both contribute to the inventory's depletion. The inventory in OW approaches zero at time t = t2. At the start of the following cycle, the client is meant to receive the shortfall quantity. Figure 1: Representation of the inventory system under warehouse system Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 1015 https://internationalpubls.com Determining the timings of t1, t2, in order to maintain the lowest feasible inventory level in terms of total relevant cost per unit of time is the aim of the suggested inventory system I2. π‘‘πΌπ‘Ÿ(𝑑) 𝑑𝑑 + πΌπ‘Ÿ(𝑑)(𝛽 + 𝑏𝑑) = βˆ’π΄π‘’π΅π‘‘, 0 ≀ 𝑑 ≀ 𝑑1 …..(1) π‘‘πΌπ‘œ(𝑑) 𝑑𝑑 + πΌπ‘œ(𝑑)(𝛼 + π‘Žπ‘‘) = 0 , 0 ≀ 𝑑 ≀ 𝑑1 ……(2) π‘‘πΌπ‘œ(𝑑) 𝑑𝑑 + πΌπ‘œ(𝑑)(𝛼 + π‘Žπ‘‘) = βˆ’π‘ƒπ‘’π‘„π‘‘, 𝑑1 ≀ 𝑑 ≀ 𝑑2 ……(3) πΌπ‘Ÿ(𝑑1) = 0, πΌπ‘œ(𝑑1) = π‘Š, πΌπ‘œ(𝑑2) = 0 {Boundary Condition} πΌπ‘Ÿ(𝑑) = 𝐴 [𝑑1 βˆ’ 𝑑 + 𝑑1 2 ( 𝐡 + 𝛼 2 ) + 𝑏𝑑1 3 2 + 𝑑2 ( 𝛼 βˆ’ 𝐡 2 ) + 𝑑3 ( 2𝑏 + 3𝐡𝛼 + 3𝛼2 6 ) + 𝑑4 ( 3𝑏𝛼 + 5𝑏𝛼 12 ) βˆ’ ( 𝑏2𝑑5 12 ) βˆ’ 𝑏𝑑𝑑1 βˆ’ 𝑑𝑑1 2 ( 𝐡𝛼 + 𝛼2 2 ) βˆ’ 𝑑2𝑑1 2 ( 𝑏𝐡 + 𝑏𝛼 4 ) βˆ’ 𝑏 2 𝑑2𝑑1 βˆ’ 𝑏2𝑑2𝑑1 3 12 βˆ’ 𝑏𝛼𝑑𝑑1 3 6 ] In the RW, the commutative inventory from 0 to 𝑑1 is provided by πΌπ‘Ÿ(0, 𝑑1) = 𝑃 ∫ [𝑑1 + 𝑑 + 𝑑1 2 ( 𝐡 + 𝛼 2 ) + 𝑏𝑑1 3 2 + 𝑑2 ( 𝛼 βˆ’ 𝐡 2 ) + 𝑑3 ( 2𝑏 + 3𝐡𝛼 + 3𝛼2 6 ) 𝑑1 0 + 𝑑4 ( 3𝑏𝛼 + 5𝑏𝛼 12 ) βˆ’ ( 𝑏2𝑑5 12 ) βˆ’ 𝑏𝑑𝑑1 βˆ’ 𝑑𝑑1 2 ( 𝐡𝛼 + 𝛼2 2 ) βˆ’ 𝑑2𝑑1 2 ( 𝑏𝐡 + 𝑏𝛼 4 ) βˆ’ 𝑏 2 𝑑2𝑑1 βˆ’ 𝑏2𝑑2𝑑1 3 12 βˆ’ 𝑏𝛼𝑑𝑑1 3 6 ]𝑑𝑑 πΌπ‘Ÿ(0, 𝑑1) = 𝑃 [ 𝑑1 2 2 + 𝑑1 3 ( 2𝐡+𝛼 6 ) + 𝑑1 4 ( 2𝑏+3𝐡𝛼+3𝛼2 24 ) βˆ’ 𝑑1 5 ( 2𝑏𝛼+5𝑏𝛼 60 ) βˆ’ 𝑏2𝑑1 6 72 ] …(4) From the equation (2), we calculated the inventory stock for the rented warehouse πΌπ‘Ÿ(𝑑) = π‘Šπ‘’βˆ’π›Ύπ‘‘βˆ’ 𝑐𝑑2 2 …(5) Similarly, we calculated the inventory stock, from the equation (3) 𝐼0(𝑑) = 𝐴 [𝑑2 βˆ’ 𝑑 + 𝑑2 2 ( 𝛾+𝐡 2 ) + 𝑐𝑑2 3 6 + 𝑑2 ( π›Ύβˆ’π΅ 2 ) + 𝑑3 ( 2𝑐+3𝐡𝛾+3𝛾2 6 ) + 𝑑4 ( 3𝑐𝐡+5𝑐𝛾 12 ) + ( 𝑐2𝑑5 12 ) βˆ’ 𝛾𝑑𝑑2 βˆ’ 𝑑𝑑2 2 ( 𝛾𝐡+𝛾2 2 ) βˆ’ 𝑑2𝑑2 2 ( 𝑐𝐡+𝑐𝛾 4 ) βˆ’ 𝑐 2 𝑑2𝑑2 βˆ’ 𝑐2𝑑2𝑑2 3 12 βˆ’ 𝑐𝛾𝑑𝑑2 3 6 ] ….(6) For the time interval[0, 𝑑2], the commutative inventory in OW is provided by 𝐼0(0, 𝑑2) = ∫ 𝐼0(𝑑)𝑑𝑑 𝑑1 0 + ∫ 𝐼0(𝑑) 𝑑2 𝑑1 𝑑𝑑 (From the equation (6)) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 1016 https://internationalpubls.com 𝐼0(0, 𝑑2) = π‘Š (𝑑1 βˆ’ 𝛾𝑑1 2 2 βˆ’ 𝑐𝑑1 3 6 ) + 𝐴 [βˆ’π‘‘1𝑑2 + ( 𝑑1 2+𝑑2 2 2 ) + ( 𝛾+𝐡 2 ) (𝑑2 3 βˆ’ 𝑑1𝑑2 2) + 𝑐 6 (𝑑2 4 βˆ’ 𝑑1𝑑2 3) + ( π›Ύβˆ’π΅ 6 ) (𝑑2 3 βˆ’ 𝑑1 3) + ( 2𝑐+3𝑐𝐡+3𝛾2 24 ) (𝑑2 4 βˆ’ 𝑑1 4) + ( 3𝑐𝐡+5𝑐𝛾 60 ) (𝑑2 5 βˆ’ 𝑑1 5) + 𝑐2 72 (𝑑2 6 βˆ’ 𝑑1 6) βˆ’ 𝛾 2 (𝑑2 3 βˆ’ 𝑑2𝑑1 2) βˆ’ ( 𝛾𝐡+𝛾2 4 ) (𝑑2 4 βˆ’ 𝑑1 2𝑑2 2) βˆ’ 𝛾𝑐 12 (𝑑2 5 βˆ’ 𝑑1 2𝑑2 3) βˆ’ 𝑐 6 (𝑑2 4 βˆ’ 𝑑2𝑑1 3) βˆ’ 𝑐2 36 (𝑑2 6 βˆ’ 𝑑1 3𝑑2 3) βˆ’ ( 𝛾𝑐+𝑐𝐡 12 ) (𝑑2 5 βˆ’ 𝑑1 3𝑑2 2)] …. (7) Continuity of𝐼0(𝑑)at time 𝑑 = 𝑑1, if follows that: 𝐼0(𝑑1) = π‘Šπ‘’βˆ’π›Ύπ‘‘1βˆ’ 𝑐𝑑1 2 2 = 𝐴 [𝑑2 βˆ’ 𝑑1 + 𝑑2 2 ( 𝛾 + 𝐡 2 ) + 𝑐𝑑2 3 6 + 𝑑1 2 ( 𝛾 βˆ’ 𝐡 2 ) + 𝑑1 3 ( 2𝑐 + 3𝐡𝛾 + 3𝛾2 6 ) + 𝑑1 4 ( 3𝑐𝐡 + 5𝑐𝛾 12 ) + ( 𝑐2𝑑1 5 12 ) βˆ’ 𝛾𝑑1𝑑2 βˆ’ 𝑑1𝑑2 2 ( 𝛾𝐡 + 𝛾2 2 ) βˆ’ 𝑑1 2𝑑2 2 ( 𝑐𝐡 + 𝑐𝛾 4 ) βˆ’ π‘Ž 2 𝑑1 2𝑑2 βˆ’ 𝑐2𝑑1 2𝑑2 3 12 βˆ’ 𝑐𝛾𝑑1𝑑2 3 6 ] π‘Š (1 βˆ’ 𝛾𝑑1 βˆ’ 𝑐𝑑1 2 2 ) = 𝐴 [ 𝑑2 βˆ’ 𝑑1 + 𝑑2 2 ( 𝛾+𝐡 2 ) + 𝑐𝑑2 3 6 + 𝑑1 2 ( π›Ύβˆ’π΅ 2 ) + 𝑑1 3 ( 2𝑐+3𝐡𝛾+3𝛾2 6 ) +𝑑1 4 ( 3𝑐𝐡+5𝑐𝛾 12 ) + ( 𝑐2𝑑1 5 12 ) βˆ’ 𝛾𝑑1𝑑2 βˆ’ 𝑑1𝑑2 2 ( 𝛾𝐡+𝛾2 2 ) βˆ’π‘‘1 2𝑑2 2 ( 𝑐𝐡+𝑐𝛾 4 ) βˆ’ 𝑐 2 𝑑1 2𝑑2 βˆ’ 𝑐2𝑑1 2𝑑2 3 12 βˆ’ 𝑐𝛾𝑑1𝑑2 3 6 ] …(8) The following items have deteriorated in both RW and OW: π·π‘Ÿ = πΌπ‘Ÿ(0) βˆ’ 𝐷𝑑1 π·π‘Ÿ = 𝐴 [ 𝑑1 2 2 + 𝑑1 3 ( 2𝐡 + 𝛼 6 ) + 𝑑1 4 ( 2𝑏 βˆ’ 3𝛼2 βˆ’ 3𝐡𝛼 24 ) βˆ’ 𝑑1 5 ( 2𝑏𝐡 + 5𝑏𝛼 60 ) βˆ’ 𝑏2𝑑1 6 72 ] βˆ’ 𝐴𝑒𝐡𝑑1𝑑1 = 𝐴 [ 𝑑1 2 2 + 𝑑1 3 ( 2𝐡 + 𝛽 6 ) + 𝑑1 4 ( 2𝑏 βˆ’ 3𝛼2 βˆ’ 3𝐡𝛼 24 ) βˆ’ 𝑑1 5 ( 2𝑏𝐡 + 5𝑏𝛼 60 ) βˆ’ 𝑏2𝑑1 6 72 βˆ’ 𝑑1(1 + 𝐡𝑑1)] = 𝐴 [βˆ’π‘‘1 + 𝑑1 2 ( 1βˆ’2𝐡 2 ) + 𝑑1 3 ( 2𝐡+𝛼 6 ) + 𝑑1 4 ( 2π‘βˆ’3𝛼2βˆ’3𝐡𝛼 24 ) βˆ’ 𝑑1 5 ( 2𝑏𝐡+5𝑏𝛼 60 ) βˆ’ 𝑏2𝑑1 6 72 ] …(9) 𝐷0 = 𝐼0(0) βˆ’ 𝐷(𝑑2βˆ’π‘‘1) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 1017 https://internationalpubls.com = π‘Š βˆ’ 𝐴𝑒𝐡(𝑑2βˆ’π‘‘1)(𝑑2βˆ’π‘‘1) = 𝐼0(𝑑1)𝑒 𝛾𝑑1+ 𝑐𝑑1 2 2 βˆ’ 𝐴(𝑑2βˆ’π‘‘1)(1 + 𝐡𝑑2βˆ’π΅π‘‘1) = 𝐴 (1 + 𝛾𝑑1 + 𝑐𝑑1 2 2 ) [𝑑2βˆ’π‘‘1 + 𝑑2 2 ( 𝛾+𝐡 2 ) + 𝑐𝑑2 3 6 + 𝑑1 2 ( π›Ύβˆ’π΅ 2 ) + 𝑑1 3 ( 2𝑐+3𝐡𝛾+3𝛾2 6 ) + 𝑑1 4 ( 3𝑐𝐡+5𝑐𝛾 12 ) + ( 𝑐2𝑑1 5 12 ) βˆ’ 𝛾𝑑1𝑑2 βˆ’ 𝑑1𝑑2 2 ( 𝛾𝐡+𝛾2 2 ) βˆ’ 𝑑1 2𝑑2 2 ( 𝑐𝐡+π‘Žπ›Ύ 4 ) βˆ’ 𝑐 2 𝑑1 2𝑑2 βˆ’ 𝑐2𝑑1 2𝑑2 3 12 βˆ’ 𝑐𝛾𝑑1𝑑2 3 6 ] βˆ’ 𝐴(𝑑2βˆ’π‘‘1)(1 + 𝐡𝑑2βˆ’π΅π‘‘1) ….(10) Total carbon emission cost 𝐢𝐸 = 𝑐1𝐹π‘₯𝑇π‘₯ ….(11) Total ordering cost for this scenario 𝐾 = 𝐾0 + 𝐾1 𝑛𝑠 … (12) As a result, the inventory system's overall average relevant cost per unit of time (including holding and deterioration costs) is as follows. 𝑇𝐢 = 1 𝑇 [Replenishment Cost + Holding Cost + Deterioration Cost + carbon emission cost] = 1 𝑇 [𝐾 + 𝐻𝑂𝐼0(π‘œ, 𝑑2) + π»π‘…πΌπ‘Ÿ(π‘œ, 𝑑1) + 𝐢𝑑(π·π‘Ÿ + 𝐷0) + 𝐢𝐸] 𝑇𝐢 = 1 𝑇 ((πΎπ‘œ + 𝐾1 𝑛𝑠) + (𝐻2 + 𝐻3 𝑛𝑠) {𝑀 (𝑑1 βˆ’ 𝛾𝑑1 2 2 βˆ’ 𝑐𝑑1 3 6 ) + 𝐴 [βˆ’π‘‘1𝑑2 + ( 𝑑1 2+𝑑2 2 2 ) + ( 𝛾+𝐡 2 ) (𝑑2 3 βˆ’ 𝑑1𝑑2 2) + 𝑐 6 (𝑑2 4 βˆ’ 𝑑1𝑑2 3) + ( π›Ύβˆ’π΅ 6 ) (𝑑2 3 βˆ’ 𝑑1 3) + ( 2𝑐+3𝐡𝛾+3𝛾2 24 ) (𝑑2 4 βˆ’ 𝑑1 4) + ( 3𝑐𝐡+5𝑐𝛾 60 ) (𝑑2 5 βˆ’ 𝑑1 5) + ( 𝑐2 72 ) (𝑑2 6 βˆ’ 𝑑1 6) βˆ’ 𝛾 2 (𝑑2 3 βˆ’ 𝑑1 2𝑑2) βˆ’ ( 𝛾𝐡+𝛾2 4 ) (𝑑2 4 βˆ’ 𝑑1 2𝑑2 2) βˆ’ 𝛾𝑐 12 (𝑑2 5 βˆ’ 𝑑1 2𝑑2 3) βˆ’ 𝑐 6 (𝑑2 4 βˆ’ 𝑑1 3𝑑2 1) βˆ’ 𝑐2 36 (𝑑2 6 βˆ’ 𝑑1 3𝑑2 3) βˆ’ ( 𝑐𝐡+𝑐𝛾 12 ) (𝑑2 5 βˆ’ 𝑑1 3𝑑2 2)]} + (𝐻1 + 𝐻2 𝑛𝑠)𝐴 { 𝑑1 2 2 + 𝑑1 3 ( 2𝐡+𝛼 6 ) + 𝑑1 4 ( 2π‘βˆ’3𝛼2βˆ’3𝐡𝛼 24 ) βˆ’ 𝑑1 5 ( 2𝑏𝐡+5𝑏𝛼 60 ) βˆ’ 𝑏2𝑑1 6 72 } + (π·π‘œ + 𝐷1 𝑛𝑠)𝐴 {βˆ’π‘‘1 + +𝑑1 2 ( 1βˆ’2𝐡 2 ) + 𝑑1 3 ( 2𝐡+𝛼 6 ) + 𝑑1 4 ( 2π΅βˆ’3𝛼2βˆ’3𝐡𝛼 24 ) βˆ’ 𝑑1 5 ( 2𝑏𝐡+5𝑏𝛼 60 ) βˆ’ 𝑏2𝑑1 6 72 } + 𝐢𝑑 {𝑃 (1 + 𝛾𝑑1 + 𝛾𝑑1 2 2 ) [𝑑2βˆ’π‘‘1 + 𝑑2 2 ( 𝛾+𝐡 2 ) + 𝑐𝑑2 3 6 + 𝑑1 2 ( π›Ύβˆ’π΅ 2 ) + 𝑑1 3 ( 2𝑐+3𝐡𝛾+3𝛾2 6 ) + 𝑑1 4 ( 3𝑐𝐡+5𝑐𝛾 12 ) + ( 𝑐2𝑑1 5 12 ) βˆ’ 𝛾𝑑1𝑑2 βˆ’ 𝑑1𝑑2 2 ( 𝛾𝐡+𝛾2 2 ) βˆ’ 𝑑1 2𝑑2 2 ( 𝑐𝐡+𝑐𝛾 4 ) βˆ’ 𝑐 2 𝑑1 2𝑑2 βˆ’ 𝑐2𝑑1 2𝑑2 3 12 βˆ’ 𝑐𝛾𝑑1𝑑2 3 6 ] βˆ’ 𝐴(𝑑2βˆ’π‘‘1) βˆ’ 𝐴𝐡(𝑑2βˆ’π‘‘1) 2}+𝑐1𝐹π‘₯𝑇π‘₯ ) …(13) 3.1 Approximate Solution Procedure: In order to minimize the total relevant cost per unit time, the following equation can be solved to determine the approximate optimal values of 𝑑1and 𝑑2, represented by 𝑑1 βˆ— and 𝑑2 βˆ—. πœ•π‘‡πΆ πœ•π‘‘1 = 0 and πœ•π‘‡πΆ πœ•π‘‘2 = 0 , and at 𝑇 = 1, πœ•2𝑇𝐢 πœ•π‘‘1 2 | (𝑑1 βˆ—, 𝑑2 βˆ—) > 0, πœ•2𝑇𝐢 πœ•π‘‘2 2 | (𝑑1 βˆ—, 𝑑2 βˆ—) > 0 and also we have to show that for the convexity of the total cost [( πœ•2𝑇𝐢 πœ•π‘‘1 2 )( πœ•2𝑇𝐢 πœ•π‘‘2 2 ) βˆ’ πœ•4𝑇𝐢 πœ•π‘‘1 2πœ•π‘‘2 2] > 0 π‘Žπ‘‘ (𝑑1 βˆ—, 𝑑2 βˆ—). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 1018 https://internationalpubls.com Now, πœ•π‘‡πΆ πœ•π‘‘1 = 0reduces to 𝑇 = 1. From the equation (13) πœ•π‘‡πΆ πœ•π‘‘1 = |(𝐻2 + 𝐻3 𝑛𝑠 ) {π‘Š (1 βˆ’ 𝛾𝑑1 βˆ’ 𝑐𝑑1 2 2 ) βˆ’ 𝐴 [𝑑2βˆ’π‘‘1 + 𝑑2 2 ( 𝛾 + 𝐡 2 ) + 𝑐𝑑2 3 6 + 𝑑1 2 ( 𝛾 βˆ’ 𝐡 2 ) + 𝑑1 3 ( 2𝑐 + 3𝐡𝛾 + 3𝛾2 6 ) + 𝑑1 4 ( 3𝑐𝐡 + 5𝑐𝛾 12 ) + ( 𝑐2𝑑1 5 12 ) βˆ’ 𝛾𝑑1𝑑2 βˆ’ 𝑑1𝑑2 2 ( 𝛾𝐡 + 𝛾2 2 ) βˆ’ 𝑑1 2𝑑2 2 ( 𝑐𝐡 + 𝑐𝛾 4 ) βˆ’ 𝑐 2 𝑑1 2𝑑2 βˆ’ 𝑐2𝑑1 2𝑑2 3 12 βˆ’ 𝑐𝛾𝑑1𝑑2 3 6 ]} + (𝐻1 + 𝐻2 𝑛𝑠 )𝑃 {𝑑1 + 𝑑1 2 ( 2𝐡 + 𝛼 2 ) + 𝑑1 3 ( 2𝑏 βˆ’ 3𝛼2 βˆ’ 3𝐡𝛼 6 ) βˆ’ 𝑑1 4 ( 2𝑏𝐡 + 5𝑏𝛼 12 ) βˆ’ 𝑏2𝑑1 5 12 } + (π·π‘œ + 𝐷1 𝑛𝑠 ) {βˆ’1 + 𝑑1(1 βˆ’ 2𝐡) + 𝑑1 2 ( 2𝐡 + 𝛼 2 ) + 𝑑1 3 ( 2𝑏 βˆ’ 3𝛼2 βˆ’ 3𝐡𝛼 6 ) βˆ’ 𝑑1 4 ( 2𝑏𝐡 + 5𝑏𝛼 12 ) βˆ’ 𝑏2𝑑1 5 12 } +𝐢𝑑 {𝑃 (1 + 𝛾𝑑1 + 𝑐𝑑1 2 2 ) [βˆ’1 + 𝑑1(𝛾 βˆ’ 𝐡) + 𝑑1 2 ( 2𝑐 + 3𝐡𝛾 + 3𝛾2 2 ) + 𝑑1 3 ( 3𝑐𝐡 + 5𝑐𝛾 3 ) + ( 5𝑐2𝑑1 4 12 ) βˆ’ 𝛼𝑑2 βˆ’ 𝑑2 2 ( 𝛾𝐡 + 𝛾2 2 ) βˆ’ 𝑑1𝑑2 2 ( 𝑐𝐡 + 𝑐𝛾 2 ) βˆ’ 𝑐 2 𝑑1 2𝑑2 βˆ’ 𝑐2𝑑1 2𝑑2 3 6 βˆ’ 𝑐𝛾𝑑2 3 6 ]} + 𝑃(𝛾 + 𝑐𝑑1) [𝑑2βˆ’π‘‘1 + 𝑑2 2 ( 𝛾 + 𝐡 2 ) + 𝑐𝑑2 3 6 + 𝑑1 2 ( 𝛾 βˆ’ 𝐡 2 ) + 𝑑1 3 ( 2𝑐 + 3𝐡𝛾 + 3𝛾2 6 ) + 𝑑1 4 ( 3𝑐𝐡 + 5𝑐𝛾 12 ) + ( 𝑐2𝑑1 5 12 ) βˆ’ 𝛾𝑑1𝑑2 βˆ’ 𝑑1𝑑2 2 ( 𝛾𝐡 + 𝛾2 2 ) βˆ’ 𝑑1 2𝑑2 2 ( 𝑐𝐡 + 𝑐𝛾 4 ) βˆ’ 𝑐 2 𝑑1 2𝑑2 βˆ’ 𝑐2𝑑1 2𝑑2 3 12 βˆ’ 𝑐𝛾𝑑1𝑑2 3 6 ] + 𝑃 + 2𝑃𝐡(𝑑2βˆ’π‘‘1)| = 0 We can determine the values of 𝑑1 denoted by 𝑑1 βˆ—, now, we have πœ•π‘‡πΆ πœ•π‘‘1 = 0 = 𝐴(𝐻2 + 𝐻3 𝑛𝑠) {βˆ’π‘‘1 + 𝑑2 + ( 𝛾+𝐡 2 ) (3𝑑2 2 βˆ’ 2𝑑1𝑑2) + 𝑐 6 (4𝑑2 3 βˆ’ 3𝑑1𝑑2 3) + ( π›Ύβˆ’π΅ 2 ) 𝑑2 2 + 𝑑2 3 ( 2𝑐+3𝐡𝛾+3𝛾2 6 ) + 𝑑2 4 ( 3𝑐𝐡+5𝑐𝛾 12 ) + ( 𝑐2𝑑2 5 12 ) βˆ’ 𝛾 2 (3𝑑2 2 βˆ’ 𝑑1 2) βˆ’ ( 𝛾𝐡+𝛾2 2 ) (4𝑑2 3 βˆ’ 2𝑑1 2𝑑2) βˆ’ 𝛾𝑐 12 (5𝑑2 4 βˆ’ 3𝑑1 2𝑑2 2) βˆ’ 𝑐 6 (4𝑑2 3 βˆ’ 𝑑1 3) βˆ’ 𝑐2 36 (6𝑑2 5 βˆ’ 3𝑑1 3𝑑2 2) βˆ’ ( 𝑐𝐡+𝑐𝛾 12 ) (5𝑑2 4 βˆ’ Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 1019 https://internationalpubls.com 2𝑑1 3𝑑2)} + 𝐢𝑑 {𝑃 (1 + 𝛾𝑑1 + 𝑐𝑑1 2 2 ) [1 + 𝑑2(𝛾 + 𝐡) + 𝑐𝑑2 2 2 βˆ’ 𝛾𝑑1 βˆ’ 𝑑1𝑑2(𝛾𝐡 + 𝛾2) βˆ’ 𝑐𝛾𝑑1𝑑2 2 2 βˆ’ 𝑐𝑑1 2 2 βˆ’ 𝑐2𝑑1 2𝑑2 2 4 βˆ’ 𝑑1 2𝑑2 ( 𝑐𝐡+𝑐𝛾 2 )] βˆ’ 𝑃 βˆ’ 2𝐴𝐡(𝑑2 + 𝑑1)}=0 3.2 Numerical Example Table 1: Model’s inventory parameters and decision variable Inventory parameters Numerical values of the inventory parameters πΎπ‘œ 400$ per order k1 100$ per order 𝐻3 0.70 $ per unit item 𝐻4 0.05 $ per unit item 𝐻1 0.35 per unit 𝐻2 0.05 per unit$, n 10 s 0.23 π·π‘œ 0.5 $ per unit item D1 0.04 per unit$, 𝛼 0.06 𝛽 0.07 𝐴 80 𝐡 90 𝑇2 1.09 year 𝑐1 0.27 𝑇π‘₯ $75 per ton CO2 𝐹π‘₯ 2.6𝑋10βˆ’3 ton CO2 𝑑1 βˆ— 0.79 year 𝑑2 βˆ— 0.89 year 𝑇𝐢(𝑑1 βˆ—, 𝑑2 βˆ—) 5672$ 3.3 Sensitivity Analysis In this part, we discussed the behaviour of the inventory input on the decision variables for the future purpose in the field of the industrial sector; Table 2: Impact of the shipment on the total inventory cost n 𝑑1 βˆ— (Year) 𝑑2 βˆ—(Year) 𝑇𝐢(𝑑1 βˆ—, 𝑑2 βˆ—) ($) 1 0.63 0.74 6503 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 1020 https://internationalpubls.com 2 0.64 0.73 6302 3 0.68 0.76 6154 4 0.75 0.84 5934 5 0.79 0.89 5672 6 0.79 0.89 5672 Table3: Impact of the learning rate on the total inventory cost 𝑠 𝑑1 βˆ— (Year) 𝑑2 βˆ— (Year) 𝑇𝐢(𝑑1 βˆ—, 𝑑2 βˆ—) ($) 0.62 0.59 0.67 6504 0.64 0.62 0.74 6309 0.67 0.65 0.78 6266 0.69 0.66 0.83 5906 0.75 0.76 0.89 5672 0.79 0.79 0.89 5672 Table4: Impact of the carbon taxation on the total inventory cost 𝑇π‘₯ 𝑑1 βˆ— (Year) 𝑑2 βˆ— (Year) 𝑇𝐢(𝑑1 βˆ—, 𝑑2 βˆ—) ($) 75 0.79 0.89 5672 76 0.75 0.83 6508 77 0.73 0.80 6876 78 0.70 0.78 7143 Figure 2: Effect of the shipment on the total cost Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 1021 https://internationalpubls.com Figure 3: Effect of the shipment on the cycle time (π’•πŸ) Figure 4: Effect of the shipment on the cycle time (π’•πŸ) Figure 5: Effect of the learning rate on the total cost 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.5 2 2.5 3 3.5 4 4.5 5 C yc le t im e (t 1) Number of shipment 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6 A xi s Ti tl e( t2 ) Number of shipment Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 1022 https://internationalpubls.com Figure 6: Effect of the learning rate on the cycle time(π’•πŸ) Figure 7: Effect of the learning rate on the cycle time(π’•πŸ) Figure 8: Effect of the carbon taxation on the total cost 0 0.2 0.4 0.6 0.8 1 0.62 0.67 0.72 0.77 C yc le t im e (t 1) Learning rate 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.62 0.67 0.72 0.77 C yc le t im e (t 2) Learning rate Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 1023 https://internationalpubls.com Figure 9: Effect of the carbon taxation on the cycle time (π’•πŸ) Figure 10: Effect of the carbon taxation on the cycle time (π’•πŸ) Observation and managerial insights ● Effect of the shipment We observed that from the table-2, if the number of shipment increases then the values of 𝑑1 βˆ— and 𝑑2 βˆ— increase but total cost decreases from 1to 5, consequently, the buyer gets more information for the exercise of shipment and also presented the Figure 2, 3 and 4. ● Effect of learning effect The role of the learning is more effective in this scenario and it minimizes the total inventory cost. From the table -3, if the learning rate increases then 𝑑1 βˆ— and 𝑑2 βˆ— increase but total inventory cost decreases and after some more shipment, it is constant due to learning in holding cost , ordering cost and deteriorating cost. The property of the deterioration of the product need to know to the seller and buyer during the leading of the business. The graphical represents in the Figure 5,6 and 7. ● Effect of carbon tax 0.68 0.7 0.72 0.74 0.76 0.78 0.8 75 75.5 76 76.5 77 77.5 78 C yc le t im e (t 1) Carbon taxation rate 0.76 0.78 0.8 0.82 0.84 0.86 0.88 0.9 74.5 75 75.5 76 76.5 77 77.5 78 78.5 C yc le t im e (t 2) Carbon Taxation Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 1024 https://internationalpubls.com From the table -4, if the carbon taxation increases then 𝑑1 βˆ— and 𝑑2 βˆ— decrease but total inventory cost increase because carbon taxation is also one type of the penalty for more carbon emits and represents in the Figure 8,9 and 10. 4. Conclusion In actuality, items that are maintained in stock may lose value or function over time. Therefore, the impact of inventory model deterioration cannot be disregarded. Demand has been considered as an exponentially growing function of time when discussing the inventory problem. The exponentially growing demand determines the replenishment rate. We first build two potential storage models in the suggested model, assuming that the demand rate is predictable. Production does not take the inventory level in OW into consideration, which is W and S of RW. After the RW level is depleted, the consumer is served by OW. Considering that deterioration is also taken into account in both warehouses. The current approach works with recently launched products because, if clients are happy with the product's quality and pricing, demand for these items rises quickly. Finally examined the effect of learning and carbon emission on the variable decisions and total inventory cost and other input variable gave positive effect on the decision variable. We might expand the models created in this paper for next study by adding more practical scenarios like multiple goods and quantity discount regulations as well as fuzzy environment. References [1] Donaldson W. A., (1977), β€œInventory Replenishment Policy for a Linear Trend in Demand An Analytical Solution”, Operational Research Quarterly, 28: 663–670. [2] Silver E. A., and Meal H. C., (1969), β€œA Simple Modification of the EOQ for the Case of a Varying Demand Rate”, Production and Inventory Management, 10(4): 52–65. [3] Silver E. A., (1979), β€œA Simple Inventory Replenishment Decision Rule for a Linear Trend in Demand”, Journal of Operational Research Society, 30: 71–75. [4] Dave U., (1989), β€œOn a Heuristic Inventory-Replenishment Rule for Items with a Linearly Increasing Demand Incorporating Shortages, Journal of the Operational Research Society, 38(5): 459–463. [5] Hill R. M., (1995), β€œOptimal EOQ Models for Deteriorating Items with Time Varying Demand”, J.O.R.S., 47: 1228–1246. [6] Hariga M. A., (1995), β€œEffects of Inflation and Time-Value of Money on an Inventory Model on an Inventory Model with Time-Dependent Demand Rate and Shortages”, E.J.O.R., 81(3): 512–520. [7] Goswami A., and Chaudhuri K. S., (1991), β€œAn EOQ Model for Deteriorating Items with a Linear Trend in Demand”, J.O.R.S., 42(12): 1105–1110. [8] Bhunia A. K., and Maiti M., (1998), β€œA Two-Warehouse Inventory Model for Deteriorating Items with a Linear Trend in Demand and Shortages”, J.O.R.S., 49(3): 287–292. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 1025 https://internationalpubls.com [9] Dye C. Y., (2002), β€œA Deteriorating Inventory Model with Stock Dependent Demand and Partial Backlogging Under Conditions of Permissible Delay in Payments”, Opsearch, 39(3&4): 189–200. [10] Mandal, M. and Maiti, M. (1997), β€œInventory model for damageable items with stock dependent demand and shortages”, Opsearch, 34(3), 155-166. [11] Khanra S., and Chaudhuri K. S., (2003), β€œA Note on an Order-Level Inventory Model for a Deteriorating Item with Time-Dependent Quadratic Demand”, C.O.R., 30, 1901–1916. [12] Balkhi Z. T., and Benkherouf L., (2004), β€œOn an Inventory Model for Deteriorating Items with Stock dependent and Time Varying Demand Rates”, Computers & Operations Research, 31, 223–240. [13] Zhou Y. W., and Yang S. L., (2005), β€œA Two-Warehouse Inventory Model for Items withStock-Level-Dependent Demand Rate”, I.J.P.E., 95(2): 215–228. [14] Hou K. L., (2006), β€œAn Inventory Model for Deteriorating Items with Stock Dependent Consumption Rate and Shortages Under Inflation and Time Discounting”, E.J.O.R., 168, 463–474. [15] Ouyang L. Y., et al., (2008), β€œRetailer’s Ordering Policy for Non-Instantaneous Deteriorating Items with Quantity Discount, Stock Dependent Demand and Stochastic Backorder Rate”, J. of Chinese Institute of Industrial Engineers, 25(1): 62–72. [16] Panda S., Saha S., and Basu M., (2007), β€œAn EOQ Model with Generalized Ramp-Type Demand and Weibull Distribution Deterioration”, Asia Pacific Journal of Operational Research, 24(1):1–17. [17]. Jayaswal, M. K., & Mittal, M. (2022),β€œImpact of Learning on the Inventory Model of Deteriorating Imperfect Quality Items with Inflation and Credit Financing under Fuzzy Environment”. International Journal of Fuzzy System Applications (IJFSA), 11(1), 1-36. [18]. Alsaedi, B. S., Alamri, O. A., Jayaswal, M. K., & Mittal, M. (2023),β€œA sustainable green supply chain model with carbon emissions for defective items under learning in a fuzzy environment”. Mathematics, 11(2), 301. [19]. Jayaswal, M. K., Mittal, M., Sangal, I., & Tripathi, J. (2021),β€œFuzzy-Based EOQ Model With Credit Financing and Backorders Under Human Learning”. International Journal of Fuzzy System Applications (IJFSA), 10(4), 14-36. [20]. Kumar, P., Yadav, V., Naik, P. J., Malik, A. K., & Alaria, S. K. (2023, June). Analysis of fuzzy inventory model with sustainable transportation. In AIP Conference Proceedings (Vol. 2782, No. 1). AIP Publishing. [21]. Malik, A.K., Yadav, S.K. and Yadav, S.R. (2012) Optimization Techniques, I. K International Pub. Pvt. Ltd., New Delhi. [22]. Singh, Y., Arya, K., & Malik, A. K. (2014). Inventory control with soft computing techniques. International Journal of Innovative Technology and Exploring Engineering, 3(8), 80-82. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 3 (2025) 1026 https://internationalpubls.com [23]. Tyagi, T., Kumar, S., Malik, A. K., & Vashisth, V. (2023). A novel neuro-optimization technique for inventory models in manufacturing sectors. Journal of Computational and Cognitive Engineering, 2(3), 204-209. [24]. Verma, P., Chaturvedi, B. K., & Malik, A. K. (2022). Comprehensive Analysis and Review of Particle Swarm Optimization Techniques and Inventory System. International Journal on Future Revolution in Computer Science & Communication Engineering, 8(3), 111-115. [25]. Yadav, S. R., & Malik, A. K. (2014). Operations research. Oxford University Press. [26]. Yadav, V., Chaturvedi, B. K., & Malik, A. K. (2022). Advantages of fuzzy techniques and applications in inventory control. International Journal on Recent Trends in Life Science and Mathematics, 9(3), 09-13.