Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 1 (2024) 177 https://internationalpubls.com An Inventory Model with Fluctuate Ordering and Holding Cost with Salvage Value for Time Sensitive Demand and Partial Backlogging Garima Khare1, Garima Sharma1 1Department of Mathematics, School of Liberal Arts and Sciences, Mody University of Science and Technology, Lakshmangarh-332311, Rajasthan, India Corresponding author: garimashish188@gmail.com Contributing author: garimasharma.slas@modyuniversity.ac.in Article History: Received: 12-10-2023 Revised: 24-11-2023 Accepted: 10-12-2023 Abstract: Inventory models are essential in evaluating a wide range of genuine situations that occur in locations, including grocery and vegetable markets, market yards, petroleum exploration businesses, etc. This article introduces an economic order quantity (EOQ) profit optimization for degrading products. It analyzes how variable ordering prices and holding charges affect profit margins within restricted planning horizons. Here demand rate is projected to be time sensitive, and the worsening price is proportional to time. Models of inventories for decaying objects are established. The dilemma is solved when shortages are considered acceptable and partially backordered. The holding and ordering expenses tend to fluctuate. The salvage value is allocated to items in the system that have deteriorated. A numerical example is used to discuss the sensitivity of the models. Further, We exemplify that the significantly reduced cost coefficient is convex if evaluated simultaneously and finds the most efficient solution. The statistical analysis reveals that a suitable policy can benefit the retailer, especially for worsening products. Keywords: Partial backlogging, Deterioration rate, salvage value, shortages, variable ordering, and holding cost. 1. Introduction Change, decay, damage, obsolescence, spoilage, and pilferage all contribute to the loss of a stock's usability or marginal value. Medicine, blood, seafood, liquor, gas, food, and radioactive substances all have a limited shelf life and begin to degrade immediately after they are received. As a result, the deterioration of the product cannot be overlooked. During stockouts, most inventory models incorrectly assume that all demand is lost or backlogged. In reality, some customers wait for replenishment, particularly if the delay is brief, while others become agitated and leave because of these stockouts, Backlogging, in part has been implemented in this investigation. Most academics believe that scarcity is entirely accumulated. Some clients choose to wait patiently for delays amid a scarcity, while others do not. The financial burden of missing out on sales should be taken into consideration in the modelling methodologies. Decaying models of stocks have received significant attention in the last couple of decades. For a constant demand, Ghare and Schrader [1] became the first to acquire a continuously depleting inventory. Through rectifying and altering the error in Shah and Jaiswal's analysis [3], in determining The mean quantity of inventory storage cost, Aggarwal [2], created an Order-specific stock model. mailto:garimashish188@gmail.com mailto:garimasharma.slas@modyuniversity.ac.in Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 1 (2024) 178 https://internationalpubls.com Goyal and Giri [4], have recently released a comprehensive review of research on deteriorating inventories. They asserted that due to the fact that numerous inventory items, such as technological goods and trendy clothing, experience demand rate swings, applying the concept of a consistent rate of demand to them is not invariably suitable. Many items experience increased necessitate throughout the expansion phase during which the item they produce endures. Alamri and Balkhi [5], investigated the appropriate measure of a production order for perishable goods with cyclical demand and also deterioration rates in connection to memory loss and learning. Dye et al. [6], estimated the optimal lot size and selling price given an exponential partial backlog and a variable rate of deterioration. According to them, the proportion of client backlogs in their purchases augments as the time between replenishments decreases. This investigation predicts a conceptual framework for EOQ inventories that accounts for decaying objects and demand that decreases exponentially. V. K. Mishra [7] developed an A stock approach that incorporates time-dependent variables storage costs, deterioration, salvage value, and scarcity. Vinod Kumar et al.[8], regarded as an inventory framework for degrading goods under a partial stockpile with holding costs that fluctuate with time and demand. The concept of quadratic demand, time-dependent deterioration without shortages, and salvage value was proposed by Mohan and R. Venkateswarlu [9]. Kavitha Priya and K. Senbagam [10] proposed a method for time-dependently deteriorating goods with quadratic time-varying demand and a partial backlog. Numerous academics developed alternative inventory models based on different patterns of demand. Pervin et al. [11] created an inventory model with time-based demand and a stochastic deterioration rate. Singh and Mishra [12] proposed a green inventory model for non-instantaneous substitutable degrading products that employ a combined ordering strategy and carbon emissions. S. Fatma, V. K. Mishra, and R. Singh [13], Model of Inventory for Immediate Deterioration Goods with Time Sensitive Demand for Post COVID-19 Recovery. This paper contains, it is anticipated that the demand rate will be time sensitive and that the rate of worsening will be proportionate to time, a model of stock for items that are degrading will be created. When shortages are acceptable and partially backordered, the problem is resolved. Costs of holding and ordering are variables. Items in the system that have deteriorated are given the salvage value. An example utilizing numbers is used to discuss the models' sensitivity. The goal of this modelling is to examine the influence of variable ordering and holding costs with salvage values on the partially backlogged problem's overall inventory cost. 2. Presumptions and annotations 2.1 Presumptions This mathematical model is described by considering the subsequent presumptions and notations. Assumptions are, 1. The demand rate 𝑓(𝑑) at times it is assumed as 𝑓(𝑑) = πœ”1 + πœ”2𝑑 ; πœ”1, πœ”2 are constants. 2. Replenishment occurs. 3. Shortages are allowed. 4. πœƒ(𝑑) = πœƒπ‘‘ is the deterioration rate. 5. Both the storage cost and purchasing cost are variable 6. Here, holding cost is time sensitive i.e., where 𝛼 > 0, 𝛽 > 0 7. Here Ordering cost is πΆπ‘œ = π‘žπ›Ύβˆ’1 where π‘ž > 0, 𝛾 > 0. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 1 (2024) 179 https://internationalpubls.com Units that have deteriorated over the period of a cycle are related with the salvage value 𝛿, 0 ≀ 𝛿 < 1. Unmet demand fulfilled have been described as partially backlogged. As consumers' waiting period, (𝑇 βˆ’ 𝑑) reduces, the percentage of backorders increases. The partial reserve rate is equals to π‘’βˆ’πœ‡(π‘‡βˆ’π‘‘). where πœ‡ is the parameter for positive backlogging. 2.2 Notations Notation Description 𝐢𝑆 πΆπ‘œ 𝐢𝐷 π‘Š 𝑆 𝑄 𝐼(𝑑) 𝑑1 𝑇 𝑇𝐢 𝛼 Ξ² 𝛿 Shortage Cost per unit of time. Ordering cost per order. Deterioration cost The maximal storage amount for every procurement period. The highest quantity of inventory The amount of the order (𝑄 = π‘Š + 𝑆). Inventory level at time t. Time when shortage start. Total length of each ordering cycle. The total price of the stock for each purchase period (0, 𝑇). Holding cost parameter. Holding cost parameter. Salvage Value 3. Model Formulation This study makes the assumption that a depreciating product will need to be replenished with variable holding and ordering costs. We provide the appropriate order quantity, 𝑄, and the ideal total cost of inventory. Fig. 1 Inventory level I(t) vs. time Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 1 (2024) 180 https://internationalpubls.com The inventory level 𝐼(𝑑) falls to zero at 𝑑 = 𝑑1 due to both demand and deterioration throughout the period [0, 𝑑1], whilst shortages occur during the period [𝑑1, 𝑇] due to demand and a portion of requirements are backlogged. 𝑑𝐼1(𝑑) 𝑑𝑑 + πœƒπ‘‘πΌ1(𝑑) = βˆ’(πœ”1 + πœ”2𝑑); 0 ≀ 𝑑 ≀ 𝑑1 (1) And during the interval [𝑑1, 𝑇] the shortage occurs, so the differential equation is given by: 𝑑𝐼2(𝑑) 𝑑𝑑 == βˆ’(πœ”1 + πœ”2𝑑)π‘’βˆ’πœ‡(π‘‡βˆ’π‘‘1); 𝑑1 ≀ 𝑑 ≀ 𝑇 (2) With the boundary conditions: 𝑑 = 0, 𝐼(0) = π‘Š, 𝑑 = 𝑑1; 𝐼(𝑑1) = 0 𝑑 = 𝑇; I(𝑇) = 𝑆 Now by solving equation (1), we get, 𝐼1(𝑑) = ( πœ”1+πœ”2𝑑 πœƒ βˆ’ πœ”2 πœƒ2) π‘’πœƒ(𝑑1βˆ’π‘‘) βˆ’ ( πœ”1+πœ”2𝑑 πœƒ βˆ’ πœ”2 πœƒ2) (3) By solving equation (2), we get: 𝐼2(𝑑) = π‘’βˆ’πœ‡(π‘‡βˆ’π‘‘1) ( πœ”1+πœ”2𝑑 πœ‡ βˆ’ πœ”2 πœ‡2) βˆ’ π‘’βˆ’πœ‡(π‘‡βˆ’π‘‘) ( πœ”1+πœ”2𝑑 πœ‡ βˆ’ πœ”2 πœ‡2) (4) Now, at 𝑑 = 0 the maximal storage amount for every period is given by 𝐼1(0) = π‘Š, 𝑑 = 0 π‘Š = 𝐼1(0) = ( πœ”1 + πœ”2𝑑 πœƒ βˆ’ πœ”2 πœƒ2 ) π‘’πœƒπ‘‘1 βˆ’ ( πœ”1 πœƒ βˆ’ πœ”2 πœƒ2 ) And at 𝑑 = 𝑇 The maximum cycle-level quadratic demand is given by 𝑑 = 𝑇, 𝐼2(𝑑) = βˆ’π‘† 𝑆 = ( πœ”1 + πœ”2𝑑 πœ‡ βˆ’ πœ”2 πœ‡2 ) βˆ’ π‘’βˆ’πœ‡(π‘‡βˆ’π‘‘1) ( πœ”1 + πœ”2𝑑 πœ‡ βˆ’ πœ”2 πœ‡2 ) Now the amount purchased per round is, 𝑄 = π‘Š + 𝑆 = ( πœ”1+πœ”2𝑑 πœƒ βˆ’ πœ”2 πœƒ2) π‘’πœƒπ‘‘1 βˆ’ ( πœ”1 πœƒ βˆ’ πœ”2 πœƒ2) + ( πœ”1+πœ”2𝑑 πœ‡ βˆ’ πœ”2 πœ‡2) βˆ’ π‘’βˆ’πœ‡(π‘‡βˆ’π‘‘1) ( πœ”1+πœ”2𝑑 πœ‡ βˆ’ πœ”2 πœ‡2) (5) The price of holding per unit time originates by, 𝐻𝐢 = ∫ (𝛼 + 𝛽𝑑)𝐼1 𝑑1 0 (𝑑) 𝐻𝐢 = 1 2πœƒ4 [{2πœƒ2(𝛽(2πœ”2𝑑1 2 + 2πœ”1𝑑1) + 𝛼(2πœ”2𝑑1 + 2πœ”1)} + πœƒ(𝛽(βˆ’6πœ”2𝑑1 βˆ’ 2πœ”1) βˆ’ 4πœ”2𝛼) + 6πœ”2𝛽)(1 βˆ’ πœƒπ‘‘1) + πœƒ2(πœ”2𝛽𝑑1 2 + 2πœ”2𝛼𝑑1) + πœƒ3(βˆ’πœ”1𝛽𝑑1 2 βˆ’ 2πœ”1𝛼𝑑1)] βˆ’ { πœ”1π›Όπœƒ4+(βˆ’πœ”1π›½βˆ’2πœ”2𝛼)πœƒ+3πœ”2𝛽 πœƒ4 } (6) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 1 (2024) 181 https://internationalpubls.com The cost of ordering per order is calculated by 𝑂𝐢 = πΆπ‘œ = π‘žπ›Ύβˆ’1 (7) Now, the deteriorating cost is given by, 𝐷𝐢 = 𝐢𝐷 [π‘Š βˆ’ ∫ 𝑓(𝑑)𝑑𝑑 𝑑1 0 ] 𝐷𝐢 = 𝐢𝐷 [( πœ”1+πœ”2𝑑1 πœƒ βˆ’ πœ”2 πœƒ2) π‘’πœƒπ‘‘1 βˆ’ ( πœ”1 πœƒ βˆ’ πœ”2 πœƒ2) βˆ’ (πœ”1𝑑1 + πœ”2𝑑1 2 2 )] (8) Shortages incur costs over time is given by 𝑆𝐢 = βˆ’πΆπ‘† ∫ 𝐼2(𝑑) 𝑇 𝑑1 𝑆𝐢 = βˆ’πΆπ‘† [( πœ”1+πœ”2𝑇 πœ‡2 βˆ’ 2πœ”2 πœ‡3 ) βˆ’ ( πœ”1+πœ”2𝑑1 πœ‡2 βˆ’ 2πœ”2 πœ‡3 ) + (𝑇 βˆ’ 𝑑1) ( πœ”1+πœ”2𝑑1 πœ‡2 βˆ’ πœ”2 πœ‡3) π‘’βˆ’πœ‡(π‘‡βˆ’π‘‘1)] (9) The salvage value πœ•, 0 ≀ πœ• < 1 𝑆𝑉 = 𝛿(𝐢𝐷) (10) Therefore, The entire cost per unit time per unit cycle is provided by, 𝑇𝐢 = 1 𝑇 (𝐻𝐢 + 𝑆𝐢 + 𝑂𝐢 + 𝐷𝐢 βˆ’ 𝑆𝑉) = 1 𝑇 [ 1 2πœƒ4 [{2πœƒ2(𝛽(2πœ”2𝑑1 2 + 2πœ”1𝑑1) + 𝛼(2πœ”2𝑑1 + 2πœ”1)} + πœƒ(𝛽(βˆ’6πœ”2𝑑1 βˆ’ 2πœ”1) βˆ’ 4πœ”2𝛼) + 6πœ”2𝛽)(1 βˆ’ πœƒπ‘‘1) + πœƒ2(πœ”2𝛽𝑑1 2 + 2πœ”2𝛼𝑑1) + πœƒ3(βˆ’πœ”1𝛽𝑑1 2 βˆ’ 2πœ”1𝛼𝑑1)] βˆ’ { πœ”1π›Όπœƒ4+(βˆ’πœ”1π›½βˆ’2πœ”2𝛼)πœƒ+3πœ”2𝛽 πœƒ4 } βˆ’ 𝐢𝑆 [( πœ”1+πœ”2𝑇 πœ‡2 βˆ’ 2πœ”2 πœ‡3 ) βˆ’ ( πœ”1+πœ”2𝑑1 πœ‡2 βˆ’ 2πœ”2 πœ‡3 ) + (𝑇 βˆ’ 𝑑1) ( πœ”1+πœ”2𝑑1 πœ‡2 βˆ’ πœ”2 πœ‡3 ) π‘’βˆ’πœ‡(π‘‡βˆ’π‘‘1)] + π‘žπ›Ύβˆ’1 + 𝐢𝐷 [( πœ”1+πœ”2𝑑1 πœƒ βˆ’ πœ”2 πœƒ2 ) π‘’πœƒπ‘‘1 βˆ’ ( πœ”1 πœƒ βˆ’ πœ”2 πœƒ2 ) βˆ’ (πœ”1𝑑1 + πœ”2𝑑1 2 2 )] βˆ’ 𝛿𝐢𝐷 [( πœ”1+πœ”2𝑑1 πœƒ βˆ’ πœ”2 πœƒ2 ) π‘’πœƒπ‘‘1 βˆ’ ( πœ”1 πœƒ βˆ’ πœ”2 πœƒ2 ) βˆ’ (πœ”1𝑑1 + πœ”2𝑑1 2 2 )]] (11) The model's objective is to locate the best values of 𝑑1 and 𝑇 to decrease the mean total expenditure per unit time 𝑇𝐢. For the optimal value, we have to get a partial derivative of 𝑑1 and 𝑇 and equating to zero. πœ•(𝑇𝐢) πœ•π‘‡ = βˆ’ 1 𝑇2 [{ 1 2 1 πœƒ4 {(πœƒ2(𝛽(2πœ”2𝑑1 2 + 2πœ”1𝑑1) + 𝛼(2πœ”2𝑑1 + 2πœ”1)) + πœƒ(𝛽(βˆ’6πœ”2𝑑1 βˆ’ 2πœ”1) βˆ’ 4πœ”2𝛼) + 6πœ”2𝛽)(βˆ’π‘‘1πœƒ + 1) + πœƒ2(πœ”2𝛽𝑑1 2 + 2π›Όπœ”2𝑑1) + πœƒ3(βˆ’πœ”1𝛽𝑑1 2 βˆ’ 2πœ”1𝛼𝑑1)} βˆ’ πœ”1π›Όπœƒ4+(βˆ’πœ”1π›½βˆ’2π›Όπœ”2)πœƒ+3πœ”2𝛽 πœƒ4 } + 𝐢𝑆 ( πœ”2𝑇+πœ”1 πœ‡2 βˆ’ 2πœ”2 πœ‡3 ) βˆ’ πœ”2𝑑+πœ”1 πœ‡2 + 2πœ”2 πœ‡3 + (𝑇 βˆ’ 𝑑1) ( πœ”2𝑑1+πœ”1 πœ‡2 βˆ’ πœ”2 πœ‡3) {1 βˆ’ πœ‡(𝑇 βˆ’ 𝑑1)} + π‘žπ›Ύβˆ’1 + 𝐢𝐷 {( πœ”2𝑑1+πœ”1 πœƒ βˆ’ πœ”2 πœƒ2) (βˆ’π‘‘1πœƒ + 1) βˆ’ πœ”1 πœƒ + πœ”2 πœƒ2 βˆ’ πœ”1𝑑1 βˆ’ 1 2 πœ”2𝑑1 2} βˆ’ 𝛿𝐢𝐷 {( πœ”2𝑑1+πœ”1 πœƒ βˆ’ πœ”2 πœƒ2 ) (βˆ’π‘‘1πœƒ + 1) βˆ’ πœ”1 πœƒ + πœ”2 πœƒ2 βˆ’ πœ”1𝑑1 βˆ’ 1 2 πœ”2𝑑1 2} + 1 𝑇 { πΆπ‘†πœ”2 πœ‡2 + ( πœ”2𝑑1+πœ”1 πœ‡2 βˆ’ πœ”2 πœ‡3) {1 βˆ’ πœ‡(𝑇 βˆ’ 𝑑1)} βˆ’ (𝑇 βˆ’ 𝑑1) + ( πœ”2𝑑1+πœ”1 πœ‡2 βˆ’ πœ”2 πœ‡3) πœ‡}] (12) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 1 (2024) 182 https://internationalpubls.com πœ•(𝑇𝐢) πœ•π‘‘1 = 1 𝑇 [ 1 2 1 πœƒ4 {(πœƒ2(𝛽4πœ”2 + 2π›½πœ”1) + 2πœ”2𝛼) βˆ’ 6πœƒπœ”2𝛽} βˆ’ (βˆ’π‘‘1πœƒ + 1) βˆ’ (πœƒ2(𝛽(2πœ”2𝑑1 2 + 2πœ”1𝑑1) + 𝛼(2πœ”2𝑑1 + 2πœ”1)) + πœƒ(𝛽(βˆ’6πœ”2𝑑1 βˆ’ 2πœ”1) βˆ’ 4πœ”1𝛼) + 6πœ”2𝛽)πœƒ + πœƒ2(2πœ”2𝛽𝑑1 + 2π›Όπœ”2) + πœƒ3(βˆ’2πœ”1𝛽𝑑1 βˆ’ 2πœ”1𝛼) βˆ’ πœ”2 πœ‡2 βˆ’ ( πœ”2𝑑1+πœ”1 πœ‡2 βˆ’ πœ”2 πœ‡3) (1 βˆ’ πœ‡(𝑇 βˆ’ 𝑑1)) + ( πœ”2𝑑1+πœ”1 πœ‡2 βˆ’ πœ”2 πœ‡3) πœ‡ + 𝐢𝐷 (( πœ”2(βˆ’π‘‘1πœƒ+1) πœƒ βˆ’ ( πœ”2𝑑1+πœ”1 πœƒ βˆ’ πœ”2 πœƒ2) πœƒ) βˆ’ πœ”1 βˆ’ πœ”2𝑑1) βˆ’ 𝛿𝐢𝐷 (( πœ”2(βˆ’π‘‘1πœƒ+1) πœƒ βˆ’ ( πœ”2𝑑1+πœ”1 πœƒ βˆ’ πœ”2 πœƒ2) πœƒ) βˆ’ πœ”1 βˆ’ πœ”2𝑑1)] (13) We get the optimal value of 𝑑1 and 𝑇 by solving the equation (12) and (13) by using MAPLE18. 4. Numerical Example A hypothetical system with the following values of multiple considerations have been investigated πœ”1 = 4, πœ”2 = 8, 𝐢𝑆 = 10, π‘ž = 7, 𝛾 = 3, 𝐢𝐷 = 3, πœƒ = 0.9, 𝛿 = 110, πœ‡ = 0.1, 𝛼 = 10, 𝛽 = 25 We utilised MAPLE 18 to solve the problem at hand the optimum rarity value, the optimal duration of the ordering cycle is 𝑇 = 32.48365353 unit time and 𝑑1 = 32.48365353per unit time. The total inventory cost is 𝑇𝐢 = 8994.779218. Fig. 2 Total inventory cost TC vs. time 5. Sensitive Analysis To investigate the impact of under and overestimating demand, deterioration, shortage cost, ordering cost, and holding cost variable on maximising system profit, sensitivity analysis was conducted on the previously described numerical example. To conduct this analysis, we varied the value of a single parameter (from -20% to +20%) while keeping all others constant. You can see the outcomes of these studies in Table 1. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 1 (2024) 183 https://internationalpubls.com Table 1 Sensitive analysis concerning different parameters Parameter % Change Change In T t1 Q TC 𝑀1 20 32.44435934 8.860404894 1195.347733 9604.449743 10 32.46402837 8.911875947 1189.898073 9597.337511 -10 32.5032319 9.014936609 1179.003918 9582.785599 -20 32.52276059 9.066524693 1173.55884 9575.351379 𝑀2 20 32.51635628 9.049409871 1410.469628 11493.10771 10 32.50150881 9.010295229 1297.459474 10541.65116 -10 32.46177675 8.906098139 1071.442485 8638.47087 -20 32.43435353 8.83455679 958.4372594 7686.673133 𝐢𝑆 20 -7.18351398 8.386866499 -1703.510229 5936.38726 10 32.67056652 9.125484922 1217.91362 9911.266027 -10 32.28341066 8.799451309 1150.266525 9271.826164 -20 32.06914071 8.633266262 1115.325694 8956.639099 q 20 32.48338612 8.963161565 1184.403494 9590.778939 10 32.48352591 8.963279287 1184.427926 9590.431993 -10 32.48376899 8.96348402 1184.470414 9589.828614 -20 32.4838723 8.963571031 1184.488472 9589.572181 𝐢𝐷 20 35.81636369 8.602033267 1384.613584 11191.63675 10 34.25891999 8.759876309 1292.017818 10427.49246 -10 30.33682651 9.247697539 1050.485579 8649.492292 -20 27.29715106 9.729613477 850.1471854 7425.210406 πœ‡ 20 -13.00795588 5.889472258 -1345.473025 -422.23059 10 32.24998362 7.911044345 1159.991768 8568.936419 -10 32.42011627 10.36323689 1159.381322 10603.69929 -20 22.38175361 14.217159 44.4843059 11651.00788 𝛿 20 35.84340917 8.599448461 1386.209366 11205.09329 10 34.27405401 8.75825374 1292.9248 10434.8029 -10 30.31444565 9.250899012 1049.061371 8640.231256 -20 16.27301748 11.63030261 -74.9144711 8121.780064 πœƒ 20 33.90269066 8.855904645 1251.776572 10065.92668 10 -10.22258989 7.315248156 -1691.350087 121.66668 -10 31.56554 9.035790259 1140.777255 9301.996899 -20 30.45273246 9.127406793 1087.892922 8973.232869 𝛾 20 32.48230768 8.962253483 1184.215025 9593.455147 10 32.4831717 8.962980991 1184.366017 9591.311112 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 1 (2024) 184 https://internationalpubls.com -10 32.48392225 8.963613108 1184.497204 9589.44817 -20 32.48407213 8.963739355 1184.523404 9589.076098 𝛼 20 32.40692585 8.970135186 1179.431191 9563.041069 10 32.44535279 8.966750509 1181.945297 9576.592845 -10 32.52182941 8.96004373 1186.946098 9603.608408 -20 32.55988174 8.956721143 1189.432992 9617.072611 𝛽 20 29.83616174 9.121449156 999.2641367 8951.810835 10 31.24618305 9.034264799 1098.401849 9282.591388 -10 33.59965309 8.903679015 1261.381806 9879.219623 -20 -11.17740669 7.293205014 -1734.118399 -287.172516 β€’ With an increase or decrease in πœ”1π‘Žπ‘›π‘‘ πœ”2, 𝑇𝐢 increases or decreases, respectively, and 𝑄 var- ies. β€’ In 𝐢𝑆 (shortage cost) and πœ‡ are small increases then 𝑇𝐢 increase but after more increase it will be decreases and if 𝐢𝑆 decreases then 𝑇𝐢 decreases. β€’ If π‘ž (Ordering variable) increases or decreases, 𝑇𝐢 not much very. β€’ If holding parameters 𝛼 and 𝛽 are increase 𝑇𝐢 decreases and if holding parameter decreases then 𝑇𝐢 increases but 𝛽 is sensitive and 𝑄 changes. β€’ If Ξ³ increases, 𝑇𝐢 increases, but when Ξ³ decreases then 𝑇𝐢 not changed and Q not very. β€’ In the small increment of πœƒ, 𝑇𝐢 and 𝑄 decreases (10%) and after that with πœƒ. TC and Q in- creases and if πœƒ is decreases both are increases β€’ 𝑇𝐢 is increase and decreases when 𝛿 and 𝐢𝑑 increases and decrease respectively. 6. Conclusions In this study, it is hypothesized that the demand rate would be time sensitive and that the rate of degradation will be proportional to time; hence, an inventory representation of things experiencing depreciation is established. The issue is remedied when shortages are tolerable and partial backorders are placed. The costs of keeping and ordering vary. The salvage value is assigned to items in the system that have degraded. The sensitivity of the models is discussed using a numerical example. 7. Application The suggested deterministic inventory model takes into consideration dynamic demand and stock, partial supply backlogs, and the expiration of perishable commodities. This strategy can be used to reduce overall inventory costs for companies in the chemical industry, where supply and demand are tightly tied and some shortages have been postponed. The model employed in this study includes a general architecture that allows variable setup costs to be accounted for while keeping stock levels constant. The cost function's convexity condition ensures the existence of a unique minimum. Inventory management is useful for non-perishable commodities such as food, gadgets, and apparel accessories. The demand function can be extended to include stochastically shifting demand patterns or stock-dependent demand rates. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 1 (2024) 185 https://internationalpubls.com 8. Future Scope The proposed model can be developed to accommodate more types of demand, such as uncertain demand, stock-dependent demand, probabilistic demand, and many others. Furthermore, in order to facilitate additional study, the model can be developed to include more realistic components such as quantity count, two-level trade credit policy, multi-item with constraint, and so on. This can be done in order to improve the model's accuracy. Furthermore, the current scope of this approach can be expanded to include less precise settings such as fuzzy, rough, random, fuzzy-random, bifuzzy, type- 2 fuzzy, and so on. References [1] Ghare, P.M., and Schrader, G.H.: A model for exponentially decaying inventory system. Inter- national Journal of Production Research, 21, 449-460 (1963) [2] Aggarwal, S.P.: A note on an order-level model for a system with constant rate of deterioration. Opsearch, 15, 184-187 (1978) [3] Shah, Y.K., and Jaiswal, M.C.: An order-level inventory model for a system with constant rate of deterioration. Opsearch, 14, 174-184 (1977) [4] Goyal S. K. and Giri B. C.: Recent trends in modeling of deteriorating inventory. 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