EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 5847 ISSN 1307-5543 – ejpam.com Published by New York Business Global A Two-Warehouse Inventory Model for Green Technology Investment: Deteriorating Items with Selling Price and Carbon Emissions Ashfar Ahmed1,2, Krishna Kummari1, Rahul Shukla3,∗ 1 Department of Mathematics, School of Science, GITAM-Hyderabad Campus, Hyderabad- 502329, India 2 Department of Mathematics, Malla Reddy (MR) Deemed to be University, Medchal-Malkajgiri, Hyderabad, Telangana - 500100, India 3 Department of Mathematical Sciences and Computing, Walter Sisulu University, Mthatha 5117, South Africa Abstract. Managing deteriorating inventory in mechatronics presents numerous opportunities. In today’s world, nearly every industry utilizes mechatronic tools and processes to slow dete- rioration, thereby reducing carbon emissions. Given the complexity of global warming, many countries are investing in various initiatives and promoting eco-friendly business practices to min- imize carbon emissions. This study examines a two-warehouse inventory model for deteriorating goods that emit carbon. Our focus is on minimizing carbon-emitting items during transportation. Reducing emissions from deteriorating inventory requires a comprehensive strategy that involves multiple supply chain partners and prioritizes environmental sustainability. By adopting green technologies, companies can effectively lower carbon dioxide emissions. In this model, demand is influenced by selling price, and partial backlogging is also considered. Additionally, incorporating time-dependent holding costs enhances the model’s applicability. The primary goal of this study is to optimize overall cycle time and costs associated with green technology investments. By optimiz- ing these factors, businesses can manage deteriorating inventory more efficiently while mitigating environmental impacts. Integrating all these elements, we propose an optimized inventory model for deteriorating goods, factoring in selling price and carbon emissions under green technology investment. The assumptions in this study suggest that the cost function is highly nonlinear, lead- ing to a constrained optimization problem. The model is solved using an algorithm implemented in Mathematica software. A numerical example is provided to illustrate the model’s application, followed by a sensitivity analysis. Finally, the optimal solution is visually represented through a graphical illustration. 2020 Mathematics Subject Classifications: 90B05, 90C30, 93A30 Key Words and Phrases: Inventory, Two-warehouses, Selling Price, Carbon emission, Green Technology Investment, Optimization ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5847 Email addresses: ahmedashfaq02@gmail.com (A. Ahmed), krishna.maths@gmail.com (K. Kummari), rshukla@wsu.ac.za (R. Shukla) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) A. Ahmed, K. Kummari, R. Shukla / Eur. J. Pure Appl. Math, 18 (2) (2025), 5847 2 of 23 1. Introduction Design and manufacturing processes play a vital role in supply chain inventory manage- ment. Perishable goods with short shelf lives must be effectively managed as part of inven- tory control for deteriorating commodities. Strategies such as First-In-First-Out (FIFO) movement, demand forecasting, and continuous monitoring help minimize waste, reduce holding costs, and maintain product quality. Businesses dealing with perishable items, such as grocery stores and pharmaceutical companies, must adhere to these practices. Many countries support initiatives aimed at reducing carbon emissions and promoting eco-friendly business policies. Sustainable manufacturing and environmentally responsible supply chains enable industries to maintain financial stability while upholding ethical standards. It is well understood that deteriorating goods contribute to lower carbon emissions. Numerous studies have been conducted to validate these approaches and offer practical insights for companies seeking to mitigate environmental impacts, including solid waste, greenhouse gas emissions, and water pollution. Benjaafar et al. [1] were the first to incorporate carbon emissions into operational decision- making related to production, inventory management, and procurement. To reduce carbon emissions, Bouchery et al. [2] developed an optimization model that integrated sustainable development with multiple objectives. Dye and Yang [3] were the first to examine how carbon emissions impact decisions on preserving or discarding deteriorating goods under various environmental regulations. Mechatronics plays a crucial role in inventory control for deteriorating goods, with its applications varying based on industry and environmental conditions. In the food sector, mechatronics can help regulate humidity and temperature, while in industrial settings, it enhances the efficiency of machines and equipment, minimizing degradation. Autonomous inventory control, data collection and analysis, climate regulation systems, automation and artificial intelligence, barcode and RFID technology, predictive main- tenance, inventory management software, quality reliability, inventory monitoring, and tracking are some notable applications [4, 5]. To optimize inventory readiness and overall costs, Kattan and Adi [6] developed an inventory model leveraging advanced technology. Raj et al. [7] proposed a hypothetical scenario utilizing green manufacturing technologies. Bhirud et al. [8] formulated a multi-objective optimization model for machining medium carbon steel. Santhi and Muthuswamy [9] explored the role of Industry 4.0 and Industry 5.0 in inventory management. Mehta et al. [10] introduced an innovative sustainable man- ufacturing strategy aimed at reducing carbon emissions. Additionally, several researchers have developed inventory models that simulate CO2 emissions, as documented in [11–22]. Various researchers have explored different carbon tax strategies for specific products in inventory models to mitigate emissions. However, product type and manufacturing processes play a more significant role in reducing carbon footprints. Zouadi et al. [23] incorporated carbon emission constraints into models for manufacturing and remanufac- A. Ahmed, K. Kummari, R. Shukla / Eur. J. Pure Appl. Math, 18 (2) (2025), 5847 3 of 23 turing. Between 2001 and 2013, Wang et al. [24] investigated energy constraints and green technology adoption in ”Chinese industrial enterprises” as part of efforts to reduce carbon emissions. Recognizing the impact of such investments, Datta [11] developed a deteriorat- ing inventory model. Mukherjee et al. [25] utilized Convolutional Neural Networks (CNN) to minimize emissions in building materials. Poswal et al. [26] proposed a demand model that accounts for both price fluctuations and stock variations. Kumar et al. [27] employed artificial intelligence and optimization techniques, while both Kumar and Gulati [28] and Kakkar et al. [29] applied different optimization strategies across various domains. Ma- hata and Debnath [30] optimized a manufacturer screening process using KKT conditions to improve efficiency. The concept of adding an additional warehouse, referred to as the rental warehouse, was first introduced by Hartley [31]. Pakkala and Achary [32] enhanced the two-warehouse inventory model by incorporating deteriorating products with a limited replenishment cycle. Bhunia and Maiti [33] developed a two-warehouse stock model based on a lin- ear consumption rate. Further research on inventory management has been conducted, as noted in studies [34–39]. Ghare and Schrader [40] were the first to investigate the impact of deterioration on inventory models. Das et al. [41] developed a model that inte- grates price-sensitive demand with declining inventory levels. Paul et al. [42] proposed an Economic Order Quantity (EOQ) model characterized by price-dependent demand and a time-dependent deterioration rate. In our model, we considered deteriorating prod- ucts that contribute to carbon emissions during storage and examined green technology investments as a strategy for lowering emissions. A shortfall in inventory management occurs when demand exceeds supply, directly im- pacting customer satisfaction and operational efficiency. One approach to addressing this issue is partially backlogging shortages, which helps balance cost efficiency and operational effectiveness. This flexible inventory model, which permits partial backlog shortages, can be applied to various inventory challenges. This study explores optimal green technology investments and their influence on company pricing decisions. To enhance realism, the model incorporates product deterioration and fluctuating holding costs. It also allows shortages, with a dynamic backlogging rate. The optimality of the total cost function is analyzed using an analytical optimization approach. Finally, Mathematica software is used to validate the proposed model through a numerical example. Here is a brief summary of the major contributions: • Before the product is received, ”n” equal installments are required. • An advance payment is necessary in a two-warehouse system. • Partial backlogging and advance payment have been taken into account. • A consistent rate of partially backlogged shortages. • The demand for the product depends upon its stock. A. Ahmed, K. Kummari, R. Shukla / Eur. J. Pure Appl. Math, 18 (2) (2025), 5847 4 of 23 • Constant rate of deterioration. The following is the sequence of the remaining sections: A discussion of assumptions and notations is included in Section 2. Section 3 describes the mathematical formulation of the model. In Section 4, by considering the different parameters, we analyze the model’s optimal solution. The paper’s conclusion is finally presented in Section 5. 2. Assumptions and Notations In the process of developing the inventory model, the subsequent assumptions were con- sidered: • The item’s demand is linearly dependent on its price, as it is represented by D(p) = ϕ1 − ϕ2p. • The market demand is influenced by the selling price. • There is no item replacement or repair. • The inventory system has an infinite planning horizon. • With zero lead time, the replenishment rate is instantaneous. • The availability of the rented warehouse (RW ) is infinite, whereas the own warehouse (OW ) has a maximum availability of W units. • In comparison to own warehouse (OW ), the costs of inventory in rented warehouse (RW ) are higher. • With ”n” evenly spaced payments, the business pays a portion ”k” of the entire cost of purchasing within the lead time ”M”. The remaining cost of purchasing is subsequently paid to acquire the lot. • Shortages are permitted, and a portion ”δ” of the demand D(p) = (ϕ1 − ϕ2p) will be backordered during the stock out time. • We assume that both warehouses will deteriorate at a steady rate ”θ” (0 < θ < 1) • To lower carbon emissions, in our approach, green technology (G) is under considera- tion for investment. It is possible to reduce carbon emissions by a certain percentage by using green technology (G), has the formula F = ρ ( 1− e−χG) where ”ρ” is the amount of ”CO2” produced, while ”χ” is the possibility of reducing ”CO2” emissions. • F = ρ ( 1− e−χG)⇒ G = − 1 χ [ ln ( 1− F ρ )] . Furthermore, when creating an inventory model, the following notation is utilised. A. Ahmed, K. Kummari, R. Shukla / Eur. J. Pure Appl. Math, 18 (2) (2025), 5847 5 of 23 Notations Units Description A $/Order Ordering Cost δ Units Backlogging unit (0 < δ < 1) S Units Level of total inventory ϕ1 Constant Demand rate’s coefficient part (ϕ1 > 0) M yr Company’s lead time for prepayments ϕ2 Constant Demand rate Constant for price (ϕ2 > 0) θ Constant Deterioration rate n Constant Equally distributed prepayments over the lead time W Units Inventory level at OW cp Units Cost per unit of purchase cd Units Cost per unit of deterioration k Constant Installment-based payment (0 < k < 1) R Units Backlogged units t1 yr RW ′s inventory level falls to zero at this point. cs Unit Shortage cost per unit cl Unit Cost of Opportunity per unit H Unit Holding costs in rupees per unit of OW F Unit Holding costs in rupees per unit of RW p Unit Selling price per unit Io(t) Unit Inventory level at any time ”t” in OW Ir(t) Unit Inventory level at any time ”t” in RW ce Constant Cost of carbon emission Rce Constant Rate of carbon emissions during deterio- ration Decision-making parameters Notations Units Description t2 yr OW ′s inventory level falls to zero at this point. G Unit Green technology investment T yr The entire length of the cycle of replenish- ment 3. Problem Definition In this section, we develop a mathematical model for a two-warehouse inventory system for deteriorating items with selling price and carbon emission under green technology investment. A. Ahmed, K. Kummari, R. Shukla / Eur. J. Pure Appl. Math, 18 (2) (2025), 5847 6 of 23 Based on the following assumptions, let’s say a company orders (S + R) units of a given product and pays a percentage ”k” of the total price in ”n” equal installments within a certain lead time ”M”. At time t = 0, the remaining purchasing cost is paid. As ”R” units are used to partially meet backlogged demand, the inventory level becomes ”S”. Now, ”W” units are preserved in ”OW”, while the remaining fraction ”S−W” is saved in ”RW”. Because ”RW” has superior facilities, the holding costs are higher than in ”OW”, causing ”RW” items to be consumed first. As a result of the constant deteriorating rate ”θ” and the need to meet customer demand ”D(p)”, ”RW”′s inventory level decreases during the time interval [0, t1]. In ”RW”, it drops to zero at t = t1. As a consequence, inventory level in ”OW” decrease due to a constant deteriorating rate ”θ” during the interval [0, t1]. Within a short time the inventory in ”OW” is depleted due to the customer’s demand ”D(p)” and deteriorating during the interval of time [t1, t2]. At t = t2, it appears to become zero. Shortages increase at a rate that remains constant ”δ” within the interval of time [t2, T ]. The following figure illustrates the inventory level during the complete cycle [0, T ] is as shown below: Figure 1: An illustration of a two-warehouse inventory model for deteriorating items. The inventory level at t = 0 to t = T is described in the differential equations as follows: dIr(t) dt + θIr(t) = −(ϕ1 − ϕ2p), 0 ≤ t ≤ t1 (1) subject to the conditions: Ir(t) = { S −W, at t = 0 0, at t = t1 (2) On evaluating the differential equations mentioned above: Ir(t) = ϕ1 − ϕ2p θ {eθ(t1−t)−1}, 0 ≤ t ≤ t1. (3) A. Ahmed, K. Kummari, R. Shukla / Eur. J. Pure Appl. Math, 18 (2) (2025), 5847 7 of 23 Additionally, at any instant in time ”t”, inventory level Io(t) in OW can be represented by the differential equations below: dIo(t) dt + θIo(t) = 0, 0 ≤ t ≤ t1 (4) dIo(t) dt + θIo(t) = −(ϕ1 − ϕ2p), t1 ≤ t ≤ t2 (5) dIo(t) dt = −δ(ϕ1 − ϕ2p), t2 ≤ t ≤ T (6) subject to the conditions: Io(t) =  W, at t = 0 0, at t = t2 −R, at t = T (7) On evaluating the differential equations mentioned above: Io(t) = We−θt, 0 ≤ t ≤ t1 (8) Io(t) = ϕ1 − ϕ2p θ {eθ(t2−t) − 1}, t1 ≤ t ≤ t2 (9) Io(t) = δ(ϕ1 − ϕ2p)(T − t)−R, t2 ≤ t ≤ T (10) In order to write the above equation, we have to consider the continuity at t = t1 and t = t2: S = W + ϕ1 − ϕ2p θ [ eθt1 − 1 ] (11) R = δ (ϕ1 − ϕ2p) (T − t2) (12) t2 = t1 + 1 θ log [ 1 + θWe−θt1 ϕ1 − ϕ2p ] (13) Presently, the following segments constitute the total cost: (a) Ordering Cost: A (b) Purchase Cost: cp(S +R) = cp ( W + ϕ1 − ϕ2p θ ( eθt1 − 1 ) + δ (ϕ1 − ϕ2p) (T − t2) ) (c) Holding Cost: F t1∫ 0 Ir (t)dt +H t1∫ 0 Io (t)dt +H t2∫ t1 Io (t)dt A. Ahmed, K. Kummari, R. Shukla / Eur. J. Pure Appl. Math, 18 (2) (2025), 5847 8 of 23 = F (ϕ1 − ϕ2p) ( θ(−t1) + eθt1 − 1 ) θ2 + H ( W −Weθ(−t1) ) θ + H(ϕ1 − ϕ2p) ( θt1 + eθ(t2−t1) − θt2 − 1 ) θ2 (d) Deterioration Cost: cdθ t1∫ 0 Ir (t)dt + cdθ t1∫ 0 Io (t)dt + cdθ t2∫ t1 Io (t)dt = cdθ(ϕ1 − ϕ2p) ( θt1 + eθ(t2−t1) − θt2 − 1 ) θ2 + cdθ ( W −Weθ(−t1) ) θ + cdθ(ϕ1 − ϕ2p) ( θ(−t1) + eθt1 − 1 ) θ2 (e) Shortage Cost: −cs T∫ t2 Io (t)dt = 1 2 csδ(T − t2) 2(ϕ1 − ϕ2p) (f) Opportunity Cost: cl (1− δ) T∫ t2 Ddt = cl (1− δ) (ϕ1 − ϕ2p) (T − t2) (g) Capital Cost: Ic [ kcp (S +R) n M n (1 + 2 + 3 + ..........+ n) ] = Ic Mkcp(n+ 1) ( (ϕ1 − ϕ2p) ( eθt1 θ − 1 θ ) + δ(ϕ1 − ϕ2p)(T − t2) +W ) 2n  (h) Carbon emission Cost: (ce.Rce) t1∫ 0 Ir (t)dt + (ce.Rce) t1∫ 0 Io (t)dt + (ce.Rce) t2∫ t1 Io (t)dt A. Ahmed, K. Kummari, R. Shukla / Eur. J. Pure Appl. Math, 18 (2) (2025), 5847 9 of 23 = (Ce.Rce) ( (ϕ1 − ϕ2p) ( θ(−t1) + eθt1 − 1 ) θ2 ) + (Ce.Rce) ( (ϕ1 − ϕ2p) ( θt1 + eθ(t2−t1) − θt2 − 1 ) θ2 ) + (Ce.Rce) ( W −Weθ(−t1) θ ) (i) Investment costs for green technology : GTIC = GT (j) Cost reductions from carbon emissions: ( 1− ρ ( 1− e−χG)) [ (ce.Rce) t1∫ 0 Ir (t)dt + (ce.Rce) t1∫ 0 Io (t)dt + (ce.Rce) t2∫ t1 Io (t)dt ] = ( 1− ρ ( 1− e−χG)) [((Ce.Rce) ( (ϕ1 − ϕ2p) ( θ(−t1) + eθt1 − 1 ) θ2 ) + (Ce.Rce) ( (ϕ1 − ϕ2p) ( θt1 + eθ(t2−t1) − θt2 − 1 ) θ2 ) + (Ce.Rce) ( W −Weθ(−t1) θ )] Consequently, the total inventory cost is TIC = 1 T  ⟨OrderingCost⟩+ ⟨PurchaseCost⟩+ ⟨HoldingCost⟩ + ⟨DeteriorationCost⟩+ ⟨ShortageCost⟩ + ⟨OpportunityCost⟩+ ⟨Capital Cost⟩ + ⟨Investment costs for green technology ⟩ + ⟨Cost reductions from carbon emissions⟩  TIC = 1 T [ (Ic Mkcp(n+ 1) ( (ϕ1 − ϕ2p) ( eθt1 θ − 1 θ ) + δ(ϕ1 − ϕ2p)(T − t2) +W ) 2n  A. Ahmed, K. Kummari, R. Shukla / Eur. J. Pure Appl. Math, 18 (2) (2025), 5847 10 of 23 + ( 1− ρ ( 1− e−χG)) [((Ce.Rce) ( (ϕ1 − ϕ2p) ( θ(−t1) + eθt1 − 1 ) θ2 ) + (Ce.Rce) ( (ϕ1 − ϕ2p) ( θt1 + eθ(t2−t1) − θt2 − 1 ) θ2 ) + (Ce.Rce) ( W −Weθ(−t1) θ )] + cp ( δ(T − t2)(ϕ1 − ϕ2p) + (ϕ1 − ϕ2p) ( eθt1 − 1 ) θ +W ) + 1 2 csδ(T − t2) 2(ϕ1 − ϕ2p) + cdθ(ϕ1 − ϕ2p) ( θt1 + eθ(t2−t1) − θt2 − 1 ) θ2 + cdθ(ϕ1 − ϕ2p) ( θ(−t1) + eθt1 − 1 ) θ2 + cl(1− δ)(ϕ1 − ϕ2p)(T − t2) + F (ϕ1 − ϕ2p) ( θ(−t1) + eθt1 − 1 ) θ2 + H(ϕ1 − ϕ2p) ( θt1 + eθ(t2−t1) − θt2 − 1 ) θ2 +A + cdθ ( W −Weθ(−t1) ) θ + H ( W −Weθ(−t1) ) θ +GT ] (14) Let t1 = ηt2, 0 < η < 1, then we get Equ.(15) from Equ.(14) TIC(t2, T ,G) = 1 T [ (Ic Mkcp(n+ 1) ( (ϕ1 − ϕ2p) ( eθηt2 θ − 1 θ ) + δ(ϕ1 − ϕ2p)(T − t2) +W ) 2n  + ( 1− ρ ( 1− e−χG)) [((Ce.Rce) ( (ϕ1 − ϕ2p) ( θ(−ηt2) + eθηt2 − 1 ) θ2 ) + (Ce.Rce) ( (ϕ1 − ϕ2p) ( θηt2 + eθ(t2−ηt2) − θt2 − 1 ) θ2 ) + (Ce.Rce) ( W −Weθ(−ηt2) θ )] + cp ( δ(T − t2)(ϕ1 − ϕ2p) + (ϕ1 − ϕ2p) ( eθηt2 − 1 ) θ +W ) + 1 2 csδ(T − t2) 2(ϕ1 − ϕ2p) A. Ahmed, K. Kummari, R. Shukla / Eur. J. Pure Appl. Math, 18 (2) (2025), 5847 11 of 23 + cdθ(ϕ1 − ϕ2p) ( θηt2 + eθ(t2−ηt2) − θt2 − 1 ) θ2 + cdθ(ϕ1 − ϕ2p) ( θ(−ηt2) + eθηt2 − 1 ) θ2 + cl(1− δ)(ϕ1 − ϕ2p)(T − t2) + F (ϕ1 − ϕ2p) ( θη(−t2) + eθηt2 − 1 ) θ2 + H(ϕ1 − ϕ2p) ( θηt2 + eθ(t2−ηt2) − θt2 − 1 ) θ2 +A + cdθ ( W −Weθη(−t2) ) θ + H ( W −Weθ(−ηt2) ) θ +GT ] (15) 4. Solution Process This section discusses the objective function’s convexity. In their research work, [38] and [43] also employed the following optimization strategy. To maximize total profit, the following requirements must be met: ∂TIC ∂t2 = 0, ∂TIC ∂T = 0, ∂TIC ∂G = 0 (16) Equation (16) yields t2, T and G’s optimal values t2 ∗, T ∗ and G∗ Following are the conditions that need to be met in order to minimize TIC(t2, T ,G) using the Hessian Matrix, a matrix of partial derivatives of second order: HM =  ∂2TIC ∂t22 ∂2TIC ∂t2∂T ∂2TIC ∂t2∂G ∂2TIC ∂T ∂t2 ∂2TIC ∂T 2 ∂2TIC ∂T ∂G ∂2TIC ∂G∂t2 ∂2TIC ∂G∂T ∂2TIC ∂G2  ∂2TIC ∂t22 > 0, ∣∣∣∣∣ ∂2TIC ∂t22 ∂2TIC ∂t2∂T ∂2TIC ∂T ∂t2 ∂2TIC ∂T 2 ∣∣∣∣∣ > 0, ∣∣∣∣∣∣∣ ∂2TIC ∂t22 ∂2TIC ∂t2∂T ∂2TIC ∂t2∂G ∂2TIC ∂T ∂t2 ∂2TIC ∂T 2 ∂2TIC ∂T ∂G ∂2TIC ∂G∂t2 ∂2TIC ∂G∂T ∂2TIC ∂G2 ∣∣∣∣∣∣∣ > 0 (17) After (17), when a matrix is positive definite, we call it a Hessian matrix HM . Based on Mathematica Software, figure 2 illustrates our complete approach for calculating the result of our presented model. A. Ahmed, K. Kummari, R. Shukla / Eur. J. Pure Appl. Math, 18 (2) (2025), 5847 12 of 23 5. Numerical Illustration This model aims to identify the optimal values t2 ∗, T ∗ and G∗ of t2, T and G in order to minimize the total inventory cost function TIC(t2, T ,G). Due to the complexity of TIC(t2, T ,G) generated in Equ.(15) it is extremely difficult to analyze it, and to determine the optimum decision variables t2, T and G. The following algorithm is implemented to solve this model. 5.1. Algorithm Figure 2: Flowchart of the solution technique for our defined model. • Set the parameters values Ic, M , cp, n, ϕ1, ϕ2, p, θ, η, δ, ρ, χ, Ce, Rce, k, W, cs, cd, F , H, A, cl. • Construct the function TIC(t2, T ,G) given by the Equ. (15). • Minimize TIC(t2, T ,G) S.T.C. 0 < t1 < t2 < T and 0 < cp < p. • Determine the optimal values of t2 ∗, T ∗, G∗ and TIC∗(t2, T ,G) 5.2. Case-1 (With an investment in Green Technology ) In this section, a numerical illustration is shown to demonstrate the model’s operations. The following values for the variables are entered as follows: Ic = 0.25, M = 0.25, cp = 10, n = 15, ϕ1 = 200, ϕ2 = 0.5, p = 15, θ = 0.3, η = 0.45, A. Ahmed, K. Kummari, R. Shukla / Eur. J. Pure Appl. Math, 18 (2) (2025), 5847 13 of 23 δ = 0.4, ρ = 2.65, χ = 2.8, Ce = 1.5, Rce = 0.3, k = 0.4, W = 100, cs = 1.5, cd = 10, F = 3, H = 1, A = 500, cl = 1.5 Here, is an optimal solution as follows: t2 ∗ = 0.8044, T ∗ = 5.5928, G∗ = 1.3179 and TIC∗ = 1507.9032 In this specific case, investment G ̸= 0 in green technology. 5.3. Case-2 (Without an investment in Green Technology ) In this section, a numerical illustration is shown to demonstrate the model’s operations. The following values for the variables are entered as follows: Ic = 0.25, M = 0.25, cp = 10, n = 15, ϕ1 = 200, ϕ2 = 0.5, p = 15, θ = 0.3, η = 0.45, δ = 0.4, ρ = 2.65, χ = 2.8, Ce = 1.5, Rce = 0.3, k = 0.4, W = 100, cs = 1.5, cd = 10, F = 3, H = 1, A = 500, cl = 1.5, G = 0 Here, is an optimal solution as follows: t2 ∗ = 0.6284, T ∗ = 6.3117 and TIC∗ = 1524.0395. In this specific case, investment G = 0 in green technology. 5.4. Sensitivity Analysis In this section, the purpose of sensitivity analysis is to determine how parameter values affect optimal values when changing them. It was assumed that one parameter would be changed by ±5% and ±10% at a time, with the other parameters remaining constant throughout the study. Results of the sensitivity analysis are shown in the following table Based on the observations from the sensitivity analysis, the following insights were drawn. • Increases in demand parameters ϕ1 and ϕ2 will result in higher demand, which will increase the total cost of the inventory. • Increasing the selling price p, causes decreased demand and related total inventory costs. • Increase in holding cost results in increase in total inventory costs. • Increase in deterioration cost θ, results in increase in total inventory costs. • Investing in green technologies reduces inventory costs when carbon emission costs per unit ce and rate Rce increases. • Due to the increase in ordering (A), purchasing (cp), shortage (cs), and opportunity (cl) costs, the total inventory cost increases. A. Ahmed, K. Kummari, R. Shukla / Eur. J. Pure Appl. Math, 18 (2) (2025), 5847 14 of 23 Parameters % change t2 ∗ T ∗ G∗ TIC∗ Ic -10% 0.7967 6.3897 1.2652 1512.4928 -5% 0.7961 6.3423 1.2412 1658.3236 +5% 0.7964 6.3899 1.2642 1799.3813 +10% 0.7966 6.3936 1.2649 1987.2740 M -10% 0.7967 6.3897 1.2652 1512.2689 -5% 0.7831 6.3721 1.2229 1722.3991 +5% 0.7925 6.3823 1.2034 1833.2162 +10% 0.7966 6.3936 1.2649 1987.2740 cp -10% 0.7967 6.3897 1.2652 1984.7456 -5% 0.7862 6.3898 1.2881 1956.4609 +5% 0.7912 6.3877 1.2786 1986.3212 +10% 0.7966 6.3936 1.2686 1987.2747 n -10% 0.7966 6.3920 1.2691 1986.1022 -5% 0.7921 6.3212 1.2421 1985.3991 +5% 0.7984 6.4121 1.2591 1985.2162 +10% 0.7943 6.3917 1.2651 1985.9432 ϕ1 -10% 0.8264 6.7490 1.2462 1813.6896 -5% 0.8123 6.8291 1.2169 1945.7824 +5% 0.8612 6.8492 1.2861 2010.3611 +10% 0.8702 6.9876 1.2821 2156.6885 ϕ2 -10% 0.7956 6.3795 1.2628 1979.4434 -5% 0.7911 6.3867 1.2690 1981.0118 +5% 0.7969 6.3991 1.2860 1988.7939 +10% 0.7977 6.4116 1.2694 1999.5885 p -10% 0.7997 6.1530 1.2820 2120.8834 -5% 0.7999 6.2521 1.3269 1984.8818 +5% 0.8654 6.4560 1.3486 1881.1123 +10% 0.7924 6.6226 1.2491 2120.3006 A. Ahmed, K. Kummari, R. Shukla / Eur. J. Pure Appl. Math, 18 (2) (2025), 5847 15 of 23 Parameters % change t2 ∗ T ∗ G∗ TIC∗ θ -10% 0.8759 6.4329 1.3135 1981.6955 -5% 0.8161 6.4516 1.3415 1984.3266 +5% 0.7999 6.4191‘ 1.3415 1987.9213 +10% 0.7279 6.3539 1.2157 1989.6414 η -10% 0.6031 5.9074 1.0346 2019.5389 -5% 0.6011 6.0021 1.0244 2006.4111 +5% 0.6121 6.2931 1.0961 2001.2162 +10% 0.5563 6.3039 1.1079 2003.4626 δ -10% 0.6881 6.6473 1.1178 1862.4281 -5% 0.6923 6.7312 1.0143 1912.2391 +5% 0.7914 6.7164 1.1986 2001.9481 +10% 0.8505 6.4046 1.2963 2009.3468 ρ -10% 0.7768 6.3825 1.2149 1987.1891 -5% 0.7814 6.3916 1.2629 1986.9191 +5% 0.8123 6.4112 1.2914 1985.2162 +10% 0.8170 6.4011 1.3120 1984.8390 W -10% 0.7801 6.1491 1.2488 1959.8834 -5% 0.7894 6.2413 1.2694 1976.8818 +5% 0.7989 6.2871 1.2786 1999.3214 +10% 0.8119 6.6253 1.2800 2011.2385 A -10% 0.7849 6.3120 1.2618 1979.8834 -5% 0.7914 6.3946 1.2769 1984.2169 +5% 0.8011 6.3969 1.2686 1987.6312 +10% 0.8082 6.4707 1.2683 1993.7895 Ce -10% 0.7634 6.4206 1.6420 1512.6434 -5% 0.7161 6.3123 1.5164 1500.2163 +5% 0.6919 6.3110 1.5206 1497.6313 +10% 0.6904 6.3004 1.5314 1400.1685 Figure 3: TIC convexity with respect to G and T A. Ahmed, K. Kummari, R. Shukla / Eur. J. Pure Appl. Math, 18 (2) (2025), 5847 16 of 23 Parameters % change t2 ∗ T ∗ G∗ TIC∗ Rce -10% 0.7864 6.2932 1.3624 1514.1223 -5% 0.7613 6.1244 1.2123 1523.6193 +5% 0.7214 6.1132‘ 1.2615 1458.6012 +10% 0.6999 6.0022 1.2425 1408.1643 F -10% 0.7912 6.1324 1.2162 1414.6321 -5% 0.7014 6.3314 1.3844 1418.7411 +5% 0.6921 6.3161 1.3012 1506.2342 +10% 0.7063 6.2141 1.2916 1510.2141 H -10% 0.7964 6.28741 1.2132 1431.4121 -5% 0.7162 6.3092 1.1469 1429.2631 +5% 0.8001 6.2648 1.2013 1506.1234 +10% 0.8000 6.2962 1.2614 1532.6432 k -10% 0.7916 6.3241 1.8322 1564.1293 -5% 0.7994 6.1268 1.9923 1582.1943 +5% 0.8012 6.2106 1.9968 1614.1023 +10% 0.7999 6.2004 2.0032 1602.3211 cs -10% 0.7964 6.3642 1.2146 1584.2316 -5% 0.8111 6.2164 1.1163 1599.1106 +5% 0.8002 6.1146 0.9911 1608.2366 +10% 0.7999 6.0246 1.0022 1610.1231 cd -10% 0.7914 6.2916 1.2783 1442.1662 -5% 0.7984 6.3100 1.2669 1486.2218 +5% 0.7916 6.3611 1.2489 1508.1162 +10% 0.8031 6.3142 1.2013 1510.1064 cl -10% 0.7610 6.2642 1.2891 1481.6123 -5% 0.7811 6.2713 1.2923 1483.1623 +5% 0.7984 6.3411 1.3216 1506.2842 +10% 0.7999 6.3942 1.3066 1510.2899 Figure 4: TIC convexity with respect to t2 and G A. Ahmed, K. Kummari, R. Shukla / Eur. J. Pure Appl. Math, 18 (2) (2025), 5847 17 of 23 Parameters % change t2 ∗ T ∗ G∗ TIC∗ χ -10% 0.7914 6.3916 1.2436 1584.4216 -5% 0.7926 6.3412 1.2416 1608.1163 +5% 0.7999 6.3816‘ 1.2812 1610.1823 +10% 0.8104 6.4102 1.2916 1600.0246 Figure 5: TIC convexity with respect to t2 and T , (G ̸= 0) A. Ahmed, K. Kummari, R. Shukla / Eur. J. Pure Appl. Math, 18 (2) (2025), 5847 18 of 23 Figure 6: TIC convexity with respect to t2 and T , (G = 0) Figure 7: TIC associated with investment in Green Technology A. Ahmed, K. Kummari, R. Shukla / Eur. J. Pure Appl. Math, 18 (2) (2025), 5847 19 of 23 Figure 8: TIC associated with selling price 6. Conclusions We developed an inventory model that is well-suited for high-demand scenarios due to its practical features. These include green technology investments to reduce carbon emissions and a prepayment facility structured in equal installments up to n times. The holding cost parameter is entirely time-dependent. Additionally, the model incorporates a two- warehouse system for storing a larger quantity of deteriorating goods. The demand rate function is influenced by both selling price and time sensitivity. In this study, we analyzed an inventory system with two storage facilities: an own ware- house (OW) and a rented warehouse (RW). Building on this framework, we optimized a two-warehouse inventory model that integrates green technology investments, deteriorat- ing items with carbon emissions, and selling price considerations. The demand rate in this model depends on selling price, and we also account for partial backlogging. The primary objective of this approach is to maximize overall cycle time and cost efficiency while investing in green technology. Optimizing these parameters enables businesses to manage deteriorating inventory more effectively while reducing the environmental impact of deterioration. The proposed model is solved using an algorithm implemented in Mathe- matica software. A numerical example is provided to demonstrate the model’s application, followed by a sensitivity analysis. Finally, a graphical illustration of the optimal solution is presented. A. Ahmed, K. Kummari, R. Shukla / Eur. J. Pure Appl. Math, 18 (2) (2025), 5847 20 of 23 This model has significant managerial implications. Business managers must recognize the importance of green marketing and integrate sustainability into their supply chain, products, and services. Implementing green technologies and sustainable policies in pro- duction, transportation, and consumption is essential for long-term societal development. Reducing carbon emissions is a key factor in driving investments in green technologies. This scenario is relevant across various industries, including food, chemical, pharmaceu- tical, automotive, agriculture, horticulture, and retail. The proposed model applies to all decomposing products that align with the given demand pattern and generate carbon emissions during storage and transportation. This model may not be applicable to all industries or product types, especially those with interval-valued or fuzzy-valued deterioration rates. Future research could explore optimizing the frequency of identical installments to minimize costs while accounting for installment-related expenses. Additionally, this model could be extended by incorporating inventory costs with interval-valued or fuzzy-valued deterioration rates. Acknowledgements The authors wish to thank the anonymous reviewers for their valuable suggestions. Conflicts of interest or competing interests The authors declare that they have no conflicts of interest. Informed Consent The authors are fully aware and satisfied with the contents of the article. References [1] S Benjaafar, Y Li, and M Daskin. 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