Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 240 https://internationalpubls.com A Two-Warehouse Inventory System with Time-Dependent Demand and Preservation Technology Puneet Kumar1, Dr. Abhinav Saxena2, Dr. Kamesh Kumar3 1,2,3Department of Mathematics, Faculty of Engineering, Teerthanker Mahaveer University, Moradabad, UP, India. E-mail- 1nirankari2608@gmail.com, 2drabhinav.engineering@tmu.ac.in, 3drkamesh.engineering@tmu.ac.in Article History: Received: 30-01-2024 Revised: 20-04-2024 Accepted: 01-05-2024 Abstract: Because of rapid environmental changes in current worldwide society, preserving technique is becoming increasingly crucial. With the aim to examine this idea, we created a model for two-warehouse inventory system that incorporates preservation technique investments. The model considers time as a linear function representing the rate of demand. Time-dependent demand is a common feature in many industries, where demand fluctuates over time due to various factors such as seasonality and market trends. The stock is kept in the restricted storage area (OW) and other warehouse (rented) is utilized for keep any excess stock that exceeds the OW's storage limit. The stock in RW is consumed first, followed by the stock in OW, as the cost of holding of the rented warehouse (RW) is more than that of the owned warehouse (OW). This paper does not allow for shortages. We use numerical simulations to illustrate the effectiveness of our proposed model under various parameters. Keywords: Inventory system, preservation technique, rented warehouse, time- dependent demand. 1. Introduction To keep long-term relationships with clients, every firm keeps things in storage. Stock exceeding the owned warehouse's storage space, which is only capable of holding a particular amount of inventory, is a common problem. In this case, the company pays a high storage cost to rent a warehouse due to good preservation techniques in the rented warehouse. The two- warehouse inventory model is a well-established approach for managing inventory in such environments, where one warehouse serves as a regular storage facility while the other acts as a backup or emergency storage. In real-world scenarios, demand for several products is not constant but changes over time due to aspects like seasonality, promotions and consumer preferences. Before storing the units in the rented warehouse, it is reasonable to keep the units in the owned warehouse first as the cost of holding of rented warehouse is comparatively greater than the cost of holding of owned warehouse, which results in reduction of inventory cost. For deteriorating items, Hartley [1] & Sarma [2] initiated to introduce the idea of model for two-warehouse. Singh & Rathore [3], Kumar et al. [4], Singh & Rathore [5], Sharma & Chaudhary [6] , Agrawal & Banerjee [7] and Singh et al. [8] considered the two-warehouse issue having shortages & backlogging. Zhou & Yang [9] examined the two-warehouse issue Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 241 https://internationalpubls.com for demand as stock dependent. Gupta et al. [10, 11] conducted an investigation on applications of warehouse and supply chain management. Saxena [12, 13] also evaluated the EOQ models for degrading items. Rastogi et al. [14] presented a model for dual-warehouse incorporating price-sensitive demand with decaying characteristics. Mandal & Phaujdar [15] also worked on dual warehouse having stock-dependent demand. Technology for product preservation is becoming increasingly crucial and necessary because of environment's continuous changes. The rate of deterioration can be reduced to a definite extent by using safety techniques and some procedural changes. In this study, we have formed an inventory control model for two-warehouse having deteriorating items by considering the facilities of preservation. Firstly, the items are consumed from rented warehouse and then consumed from owned warehouse owing to the higher holding cost of RW than owned warehouse. The demand’s rate is taken as time dependent. However, shortages aren’t allowed. We have also considered a linear function of demand. The primary focus of formulation of this model aims to reduce the total inventory cost. In the final, numerical case is given to verify the model. 2. Notations Z (I(t)): Instantaneous time dependent rate of demand; q: Quantity of Order; 𝛳: constant rate of deterioration; m(πœ‰): (= (1-e-aπœ‰)) reduction in deterioration rate πœ‰>0; πœŽπ›³: Aggregate rate of deterioration 𝜎 = (𝛳- m (πœ‰)); O: Cost of ordering; H1: Cost of holding of Rented Warehouse (RW); H2: Cost of holding of Owned Warehouse (OW); x: Cost of unit purchasing; So: Amount of Inventory in Own Warehouse; S1: Amount of inventory in Rented Warehouse; tr: The period of time in which RW inventory level drops to zero; to: The period of time in which OW inventory level drops to zero; T: Cycle length; I1(t): Stock level in RW during [0, tr]; I2(t): Stock level in OW during [0, tr]; I3(t): Stock level in OW during [tr, to]; TRC: Total relevant cost; Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 242 https://internationalpubls.com PC: Purchasing Cost; HC: Holding Cost; 3. Assumptions β€’ Rate of demand is time dependent & taken as 𝑍(𝐼(𝑑)) = π‘Ž + 𝑏𝑑 β€’ No shortages are allowed. β€’ Time horizon is taken as infinite. β€’ Preservation technique is used for reducing deterioration. β€’ Lead time is assumed as zero and replenishment rate is taken infinite. β€’ The owned warehouse (OW) has only W2 units of space, and unlimited amount of space in rented warehouse. β€’ Because the amount of holding (H1) of RW is greater than cost of holding (H2) of OW, utilization of inventory in OW begins only when the inventory in RW reaches zero. β€’ The price of transportation and time for both RW and OW are lesser. 4. Formulation of Mathematical Model The inventory is S at t = 0, with S2 units kept in OW & rest of S1 units kept in RW. Because of demand & deterioration the inventory of rented warehouse decreases in the interval [o, tr] and become zero at t = tr. After time tr, the demand for items is met by using OW inventory at [tr; to]. The depletion in level of inventory in rented warehouse is defined via the equation: 𝑑(𝐼1(𝑑)) 𝑑𝑑 + πœŽπ›³πΌ1(𝑑) = βˆ’(π‘Ž + 𝑏𝑑); 0 ≀ 𝑑 ≀ π‘‘π‘Ÿ (1) And working of inventory in Owned warehouse is give as follows: 𝑑(𝐼2(𝑑)) 𝑑𝑑 + πœŽπœƒπΌ2(𝑑) = 0; 0 ≀ 𝑑 ≀ π‘‘π‘Ÿ (2) 𝑑(𝐼3(𝑑)) 𝑑𝑑 + πœŽπœƒπΌ3(𝑑) = βˆ’(π‘Ž + 𝑏𝑑); π‘‘π‘Ÿ ≀ 𝑑 ≀ π‘‘π‘œ (3) Now solving the equation (1), (2) & (3), we get 𝐼1 = βˆ’ π‘Ž πœŽπœƒ βˆ’ 𝑏𝑑 πœŽπœƒ + 𝑏 πœŽπœƒ 2 + 𝑐1𝑒 βˆ’πœŽπœƒπ‘‘ (4) 𝐼2 = π‘’βˆ’πœŽπœƒπ‘‘π‘2 (5) 𝐼3 = βˆ’ π‘Ž πœŽπœƒ βˆ’ 𝑏𝑑 πœŽπœƒ + 𝑏 πœŽπœƒ 2 + 𝑐3𝑒 βˆ’πœŽπœƒπ‘‘ (6) Now using the boundary conditions, 𝐼1(𝑑 = 0) = 𝑆1, 𝐼1(𝑑 = π‘‘π‘Ÿ) = 0, 𝐼2(𝑑 = 0) = 𝑆2, 𝐼2(𝑑 = π‘‘π‘Ÿ) = 𝐼3(𝑑 = π‘‘π‘Ÿ), 𝐼3(𝑑 = π‘‘π‘œ = 𝑇) = 0 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 243 https://internationalpubls.com 𝐼1 = 𝑆1𝑒 βˆ’πœŽπœƒπ‘‘ + π‘Ž πœŽπœƒ (π‘’βˆ’πœŽπœƒπ‘‘ βˆ’ 1) βˆ’ 𝑏 πœŽπœƒ 2 (π‘’βˆ’πœŽπœƒπ‘‘ βˆ’ 1) βˆ’ 𝑏𝑑 πœŽπœƒ (7) 𝐼2 = π‘’βˆ’πœŽπœƒπ‘‘π‘†2 (8) 𝐼3 = βˆ’ π‘Ž πœŽπœƒ βˆ’ 𝑏𝑑 πœŽπœƒ + 𝑏 πœŽπœƒ 2 + 𝑆2𝑒 βˆ’πœŽπœƒπ‘‘ + π‘’βˆ’πœŽπœƒ(𝑑+π‘‘π‘Ÿ) [ π‘Ž πœŽπœƒ + π‘π‘‘π‘Ÿ πœŽπœƒ βˆ’ 𝑏 πœŽπœƒ 2] (9) 𝑆1 = [βˆ’ π‘Ž πœŽπœƒ + 𝑏 πœŽπœƒ 2] (1 βˆ’ π‘’πœŽπœƒπ‘‘π‘Ÿ) + π‘π‘‘π‘Ÿ πœŽπœƒ π‘’πœŽπœƒπ‘‘π‘Ÿ (10) 𝑆2 = π‘’πœŽπœƒπ‘‡ [ π‘Ž πœŽπœƒ + 𝑏𝑇 πœŽπœƒ βˆ’ 𝑏 πœŽπœƒ 2] βˆ’ π‘’πœŽπœƒπ‘‘π‘Ÿ [ π‘Ž πœŽπœƒ + π‘π‘‘π‘Ÿ πœŽπœƒ βˆ’ 𝑏 πœŽπœƒ 2] (11) Total Order Quantity π‘ž = 𝑆1 + 𝑆2 π‘ž = [βˆ’ π‘Ž πœŽπœƒ + 𝑏 πœŽπœƒ 2] (1 βˆ’ π‘’πœŽπœƒπ‘‘π‘Ÿ) + π‘π‘‘π‘Ÿ πœŽπœƒ π‘’πœŽπœƒπ‘‘π‘Ÿ + π‘’πœŽπœƒπ‘‡ [ π‘Ž πœŽπœƒ + 𝑏𝑇 πœŽπœƒ βˆ’ 𝑏 πœŽπœƒ 2] βˆ’ π‘’πœŽπœƒπ‘‘π‘Ÿ [ π‘Ž πœŽπœƒ + π‘π‘‘π‘Ÿ πœŽπœƒ βˆ’ 𝑏 πœŽπœƒ 2] (12) The following cost parameters are part of the overall relevant cost: 1. Cost of ordering 𝑂C = 𝑂𝑐 2. The amount of Carrying Cost is obtained by: 𝐻𝐢 = 𝐻1 [∫ 𝐼1 π‘‘π‘Ÿ 0 (𝑑)𝑑𝑑] + 𝐻2 [∫ 𝐼2(𝑑)𝑑𝑑 π‘‘π‘Ÿ 0 + ∫ 𝐼3(𝑑) 𝑇 π‘‘π‘Ÿ 𝑑𝑑] 𝐻𝐢 = 𝐻1 [(1 βˆ’ π‘’βˆ’πœŽπœƒπ‘‘π‘Ÿ) [ 𝑆1 πœŽπœƒ + π‘Ž πœŽπœƒ 2 βˆ’ 𝑏 πœŽπœƒ 3] βˆ’ π‘Žπ‘‘π‘Ÿ πœŽπœƒ + π‘π‘‘π‘Ÿ πœŽπœƒ 2 βˆ’ π‘π‘‘π‘Ÿ 2 2πœŽπœƒ ] + 𝐻2 [[ 𝑆2 πœŽπœƒ (1 βˆ’ π‘’βˆ’πœŽπœƒπ‘‘π‘Ÿ)] + π‘Ž πœŽπœƒ (π‘‘π‘Ÿ βˆ’ 𝑇) + 𝑏 2πœŽπœƒ (π‘‘π‘Ÿ 2 βˆ’ 𝑇2) βˆ’ 𝑏 πœŽπœƒ 2 (π‘‘π‘Ÿ βˆ’ 𝑇) + 𝑆2 πœŽπœƒ (π‘’βˆ’πœŽπœƒπ‘‘π‘Ÿ βˆ’ π‘’βˆ’πœŽπœƒπ‘‡) + [ π‘Ž πœŽπœƒ 2 + 𝑏𝑇 πœŽπœƒ 2 βˆ’ 𝑏 πœŽπœƒ 3] (π‘’βˆ’πœŽπœƒ(π‘‘π‘Ÿ+𝑇) βˆ’ π‘’βˆ’2πœŽπœƒπ‘‡)](13) Now, the complete relevant cost is given by: 𝑇𝑅𝐢 = 1 𝑇 [𝑂𝐢 + 𝐻𝐢] 𝑇𝑅𝐢 = 1 𝑇 [O + 𝐻1 [(1 βˆ’ π‘’βˆ’πœŽπœƒπ‘‘π‘Ÿ) [ 𝑆1 πœŽπœƒ + π‘Ž πœŽπœƒ 2 βˆ’ 𝑏 πœŽπœƒ 3] βˆ’ π‘Žπ‘‘π‘Ÿ πœŽπœƒ + π‘π‘‘π‘Ÿ πœŽπœƒ 2 βˆ’ π‘π‘‘π‘Ÿ 2 2πœŽπœƒ ] + 𝐻2 [[ 𝑆2 πœŽπœƒ (1 βˆ’ π‘’βˆ’πœŽπœƒπ‘‘π‘Ÿ)] + π‘Ž πœŽπœƒ (π‘‘π‘Ÿ βˆ’ 𝑇) + 𝑏 2πœŽπœƒ (π‘‘π‘Ÿ 2 βˆ’ 𝑇2) βˆ’ 𝑏 πœŽπœƒ 2 (π‘‘π‘Ÿ βˆ’ 𝑇) + 𝑆2 πœŽπœƒ (π‘’βˆ’πœŽπœƒπ‘‘π‘Ÿ βˆ’ π‘’βˆ’πœŽπœƒπ‘‡) + [ π‘Ž πœŽπœƒ 2 + 𝑏𝑇 πœŽπœƒ 2 βˆ’ 𝑏 πœŽπœƒ 3] (π‘’βˆ’πœŽπœƒ(π‘‘π‘Ÿ+𝑇) βˆ’ π‘’βˆ’2πœŽπœƒπ‘‡)]](14) Next, we will differentiate the equation (14) with respect to 𝑑1, πœ‰ π‘Žπ‘›π‘‘ 𝑇, to reduce the overall relevant cost. Required conditions for ideal values are: πœ•π‘‡π‘…πΆ(π‘‘π‘Ÿ , πœ‰, 𝑇) πœ•π‘‘π‘Ÿ = 0, πœ•π‘‡π‘…πΆ(π‘‘π‘Ÿ , πœ‰, 𝑇) πœ•πœ‰ = 0, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 244 https://internationalpubls.com πœ•π‘‡π‘…πΆ(π‘‘π‘Ÿ , πœ‰, 𝑇) πœ•π‘‡ = 0, These values give the determinant value of principal minor of hessian matrix and det(H1)> 0, det(H2)> 0, det(H3) >0. H1, H2 & H3 are the principal minor of the Hessian Matrix. The total function as a Hessian matrix is as follows: [ πœ•2𝑇𝑅𝐢 πœ•π‘‘π‘Ÿ2 πœ•2𝑇𝑅𝐢 πœ•π‘‘π‘Ÿπœ•πœ‰ πœ•2𝑇𝑅𝐢 πœ•π‘‘π‘Ÿπœ•π‘‡ πœ•2𝑇𝑅𝐢 πœ•πœ‰πœ•π‘‘π‘Ÿ πœ•2𝑇𝑅𝐢 πœ•πœ‰2 πœ•2𝑇𝑅𝐢 πœ•π‘‡πœ•πœ‰ πœ•2𝑇𝑅𝐢 πœ•π‘‡πœ•π‘‘π‘Ÿ πœ•2𝑇𝑅𝐢 πœ•πœ‰πœ•π‘‡ πœ•2𝑇𝑅𝐢 πœ•π‘‡2 ] From equation (14) we get πœ•π‘‡π‘…πΆ(π‘‘π‘Ÿ,πœ‰,𝑇) πœ•π‘‘π‘Ÿ = 1 𝑇 [𝐻1 [𝑆1𝑒 βˆ’πœŽπœƒπ‘‘π‘Ÿ + π‘Ž πœŽπœƒ (π‘’βˆ’πœŽπœƒπ‘‘π‘Ÿ) + 𝑏 πœŽπœƒ 2 (π‘’βˆ’πœŽπœƒπ‘‘π‘Ÿ βˆ’ 1) βˆ’ π‘π‘‘π‘Ÿ πœŽπœƒ ] + 𝐻2 [ π‘Ž πœŽπœƒ + π‘π‘‘π‘Ÿ πœŽπœƒ βˆ’ 𝑏 πœŽπœƒ 2 βˆ’ π‘’βˆ’πœŽπœƒ(π‘‘π‘Ÿ+𝑇) [ π‘Ž πœŽπœƒ + 𝑏𝑇 πœŽπœƒ βˆ’ 𝑏 πœŽπœƒ 2]]](15) πœ•π‘‡π‘…πΆ(π‘‘π‘Ÿ,πœ‰,𝑇) πœ•π‘‡ = βˆ’ 1 𝑇2 [O + 𝐻1 [(1 βˆ’ π‘’βˆ’πœŽπœƒπ‘‘π‘Ÿ) [ 𝑆1 πœŽπœƒ + π‘Ž πœŽπœƒ 2 βˆ’ 𝑏 πœŽπœƒ 3] βˆ’ π‘Žπ‘‘π‘Ÿ πœŽπœƒ + π‘π‘‘π‘Ÿ πœŽπœƒ 2 βˆ’ π‘π‘‘π‘Ÿ 2 2πœŽπœƒ ]] + 𝐻2 [βˆ’ 𝑆2 𝑇2πœŽπœƒ (1 βˆ’ π‘’βˆ’πœŽπœƒπ‘‘π‘Ÿ) βˆ’ π‘Žπ‘‘π‘Ÿ πœŽπœƒπ‘‡2 βˆ’ 𝑏 2πœŽπœƒ ( π‘‘π‘Ÿ 2 𝑇2 + 1) + π‘π‘‘π‘Ÿ πœŽπœƒ 2𝑇2 βˆ’ 𝑆2 𝑇2πœŽπœƒ 2 (π‘’βˆ’πœŽπœƒπ‘‘π‘Ÿ βˆ’ π‘’βˆ’πœŽπœƒπ‘‡) + 𝑆2π‘’βˆ’πœŽπœƒπ‘‡ π‘‡πœŽπœƒ + 𝑏 πœŽπœƒ 2 (π‘’βˆ’πœŽπœƒ(π‘‘π‘Ÿ+𝑇) βˆ’ π‘’βˆ’2πœŽπœƒπ‘‡) + [ π‘Ž πœŽπœƒ + 𝑏𝑇 πœŽπœƒ βˆ’ 𝑏 πœŽπœƒ 2] (2π‘’βˆ’2πœŽπœƒπ‘‡ βˆ’ π‘’βˆ’πœŽπœƒ(π‘‘π‘Ÿ+𝑇))] = 0 (16) And find also, πœ•2𝑇𝑅𝐢 πœ•π‘‘π‘Ÿ 2 = 1 𝑇 [𝐻1 [βˆ’π‘†1πœŽπœƒπ‘’βˆ’πœŽπœƒπ‘‘π‘Ÿ βˆ’ 𝑏 πœŽπœƒ [π‘’βˆ’πœŽπœƒπ‘‘π‘Ÿ + 1]] + 𝐻2 [βˆ’π‘†2πœŽπœƒπ‘’βˆ’πœŽπœƒπ‘‘π‘Ÿ + 𝑏 πœŽπœƒ + π‘’βˆ’2πœŽπœƒπ‘‘π‘Ÿ(4π‘Ž + 4π‘π‘‘π‘Ÿ) βˆ’ π‘’βˆ’πœŽπœƒ(π‘‘π‘Ÿ+𝑇) [π‘Ž + 𝑏 βˆ’ 𝑏 πœŽπœƒ ]]] (17) πœ•2𝑇𝑅𝐢 πœ•π‘‡2 = βˆ’ 1 𝑇2 [O + 𝐻1 [βˆ’π‘†1 π‘’βˆ’πœŽπœƒπ‘‘π‘Ÿ πœŽπœƒ βˆ’ π‘Ž πœŽπœƒ ( π‘’βˆ’πœŽπœƒ πœŽπœƒ βˆ’ π‘‘π‘Ÿ) + 𝑏 πœŽπœƒ 2 ( π‘’βˆ’πœŽπœƒπ‘‘π‘Ÿ πœŽπœƒ βˆ’ π‘‘π‘Ÿ) βˆ’ π‘π‘‘π‘Ÿ 2 2πœŽπœƒ ]] + 𝐻2 [ βˆ’2 𝑇3πœŽπœƒ (𝑆2𝑒 βˆ’πœŽπœƒπ‘‘π‘Ÿ + π‘Žπ‘‘π‘Ÿ) + π‘π‘‘π‘Ÿ 𝑇3πœŽπœƒ 2 (π‘‘π‘Ÿ βˆ’ 2) βˆ’ π‘’βˆ’πœŽπœƒπ‘‘π‘Ÿ πœŽπœƒ 2 [π‘Ž + π‘π‘‘π‘Ÿ + 𝑏 πœŽπœƒ ] [ 2 𝑇3 (π‘’βˆ’πœŽπœƒπ‘‘π‘Ÿ βˆ’ π‘’βˆ’πœŽπœƒπ‘‡) βˆ’ πœŽπœƒπ‘’βˆ’πœŽπœƒπ‘‡ 𝑇 [ 2 𝑇 βˆ’ πœŽπœƒ]]] (18) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 245 https://internationalpubls.com πœ•2𝑇𝑅𝐢 πœ•π‘‘π‘Ÿπœ•π‘‡ = πœ•2𝑇𝑅𝐢 πœ•π‘‡πœ•π‘‘π‘Ÿ = βˆ’ 1 𝑇2 [𝐻1 [𝑆1𝑒 βˆ’πœŽπœƒπ‘‘π‘Ÿ + π‘Ž πœŽπœƒ + 𝑏 πœŽπœƒ 2 (βˆ’π‘’πœŽπœƒπ‘‘π‘Ÿ βˆ’ 1) βˆ’ π‘π‘‘π‘Ÿ πœŽπœƒ ] + 𝐻2 [βˆ’ 1 𝑇2 [𝑆2𝑒 βˆ’πœŽπœƒπ‘‘π‘Ÿ βˆ’ π‘Ž πœŽπœƒ + π‘π‘‘π‘Ÿ πœŽπœƒ βˆ’ 𝑏 πœŽπœƒ 2 + π‘’βˆ’2πœŽπœƒπ‘‘π‘Ÿ [βˆ’ 2π‘Ž πœŽπœƒ βˆ’ 2π‘π‘‘π‘Ÿ πœŽπœƒ βˆ’ 𝑏 πœŽπœƒ 2] βˆ’ π‘’βˆ’πœŽπœƒ(π‘‘π‘Ÿ+𝑇) 𝑇 [πœŽπœƒ + 1 𝑇 ]]]] (19) 5. Effect of Preservation Technology Within this framework, we have included preservation technology in order to reduce the deterioration rate. We have taken the following values π‘Ž = 3; ΞΎ = 0.0136073; ΞΈ = 0.09 We get the value of m(ΞΎ) = 0.04 & πœŽπœƒ = 0.05 6. Numerical Illustration In order to demonstrate the model, we have taken the following inventory system: 𝑂 = 600; π‘Ž = 250; 𝑏 = 25; 𝐻1 = 0.75; 𝐻2 = 0.4; 𝑆1 = 500; 𝑆2 = 300; πœŽπœƒ = 0.05 We observed the optimal solution as π‘‘π‘Ÿ βˆ— = 2.97148; π‘‡βˆ— = 10; π‘‡π‘…πΆβˆ— = 665.0132 7. Sensitivity Analysis To analyse the sensitivity of the model, we execute a sensitivity analysis by changing the values of various key parameters such as demand parameters & deterioration rate. Table 1, Table 2 & Table 3 depicts the impact of parameter changes. Table 1: Sensitivity analysis of optimal solution with respect to variation in parameter β€˜a’ t T TRC Variation in Parameter β€˜a’ 220 2.03169 11.2 925.9562 235 2.50437 10.6 793.9031 265 3.43325 9.4 531.157011 Table 2: Sensitivity analysis of optimal solution with respect to variation in parameter β€˜b’ t T TRC Variation in Parameter β€˜b’ 20 4.97261 7.5 144.115 30 1.57538 11.6 1207.225 35 0.543249 12.8 1781.7228 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 246 https://internationalpubls.com Table 3: Sensitivity analysis of optimal solution with respect to variation in parameter β€˜πˆπœ½β€™ because of modification in the value of m(πœ‰) t T TRC Fluctuation in the value of 𝛏 Fluctuation in the value of β€˜m(πœ‰)’ Variation in value of β€˜πˆπœ½β€™ 0.0190654 0.05 0.04 1.50836 15 1501.65025 0.010735 0.03 0.06 3.90582 6.66678 238.0357 β€’ From table 1 we can see that on increasing the value of β€˜a’, the value of β€˜t’ and β€˜TRC’ decreases and the value of β€˜T’ increases. β€’ From table 2 we can see that on increasing the value of β€˜b’, the value of β€˜T’ and β€˜TRC’ increases and the value of β€˜t’ decreases. β€’ From table 3 we can see that as the value of β€˜ΞΎβ€™ decreases then value of β€˜πœŽπœƒβ€™ increases which results in increase in the value of β€˜t’ and decrease in the value of β€˜T’ and β€˜TRC’. 8. Conclusion For the dual warehouse model for inventory having demand as time-dependent and preservation technology, we found that the optimal strategy involves dynamically allocating inventory among the warehouses based on demand patterns and the effectiveness of preservation technology. We can see through the numerical illustration and sensitivity analysis that how the value of total relevant cost fluctuates for the different values of parameters. We can also see that by using the preservation technology we can reduce the rate of deterioration and preserves the products for the longer time. We have also drawn the graphs showing the variation in total cost with respect to the demand parameters. This approach maximizes the service levels while reducing the costs, ensuring that products will be available when needed Fig (A) Fig (B) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 2 (2024) 247 https://internationalpubls.com with minimizing the risk of obsolescence. 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