Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 303 https://internationalpubls.com Optimizing Inventory Management for Perishable Goods: Managing Exponential Demand Variations, Stable Holding Charges, and Partial Backlog Handling S.Ramya1 and D.Sivakumar2 1 Department of Mathematics, Sri Vasavi College, Erode, Tamil Nadu, India. Email : sramyavasavi@gmail.com 2 Department of Mathematics, Kongu Arts and Science College, Erode, Tamil Nadu, India. Email : profsiva75@gmail.com Article History: Received: 14-04-2024 Revised: 18-05-2024 Accepted: 15-06-2024 Abstract: In the ever-evolving landscape of modern business, adept inventory management stands as a cornerstone of competitiveness and profitability. This research unveils a tailored inventory optimization model designed for perishable goods, accounting for critical factors including exponential time-varying demand, consistent holding costs, and partial backlog management. The model's objective is to find an equilibrium, optimizing inventory expenses while maintaining sufficient stock to fulfil customer needs. By incorporating these key elements, businesses can enhance their inventory management strategies to adapt to changing market conditions and improve overall operational efficiency. Through analytical insights and numerical simulations, this research provides valuable guidance for decision-makers in optimizing inventory policies for perishable goods. Keywords: Perishable goods, exponential demand variations, stable holding charge, Shortages and partial backlog. 1. Introduction Efficient inventory management is fundamental for businesses operating in various industries to meet customer demands while minimizing costs and maximizing profitability. In particular, the management of deteriorating items presents unique challenges due to factors such as time-varying demand patterns and the potential for inventory deterioration over time. This paper focuses on developing an inventory optimization model specifically tailored to address these challenges, with a particular emphasis on items subject to exponential time-varying demand, stable holding costs, and partial backlog management. Deteriorating items, such as perishable goods or products with limited shelf lives, require careful handling to avoid obsolescence and minimize wastage. Moreover, the demand for such items often fluctuates over time, influenced by factors such as seasonality, market trends, and promotional activities. Conventional inventory models might overlook these dynamics, potentially resulting in suboptimal inventory levels and higher expenses. In addition to demand variability, the management Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 304 https://internationalpubls.com of deteriorating items must also consider the cost implications of holding inventory over time. Stable holding costs, representing the expenses associated with storing and maintaining inventory, play a significant role in determining the optimal inventory policy. Balancing these costs with the need to fulfil customer orders in a timely manner is essential for achieving operational efficiency and customer satisfaction. Furthermore, the presence of partial backlog management introduces another layer of complexity to the inventory optimization problem. Partial backlogging occurs when customer demand exceeds available inventory, resulting in unfulfilled orders that may be partially satisfied in subsequent periods. Effective management of partial backlogs requires careful consideration of order fulfilment priorities and inventory replenishment strategies to minimize stockouts and associated costs. Against this backdrop, this paper proposes an inventory optimization model that integrates these key factors to provide decision-makers with actionable insights for managing deteriorating items effectively. By capturing the interplay between exponential time-varying demand, stable holding costs, and partial backlog management, the proposed model offers a comprehensive framework for optimizing inventory policies and enhancing overall supply chain performance. Through empirical validation and numerical experiments, we demonstrate the practical applicability and the efficacy of the suggested model in real-world inventory management scenarios. 2. Literature Overview The literature on inventory optimization for deteriorating things through various demand patterns, holding charge structures, also backlog management strategies is rich and diverse. Several notable studies have contributed to this field, offering insights into the development of effective inventory models tailored to specific contexts. Here, we review relevant literature that addresses the complexities of managing perishable goods, encompassing aspects such as time-varying demand, stable holding charges, also partial backlog management. Su, Lin, and Tsai (1999) investigated a deterministic model for inventory management of perishable goods experiencing an exponential decline in demand. Their study aimed to provide insights into optimal inventory policies under deteriorating conditions, considering the impact of demand decline on production and inventory management decisions. Shah and Shah (2000) carried out an extensive literature review on inventory models for perishable goods, emphasizing important trends, methodologies, and research gaps within this domain. Their analysis provided valuable insights into the progression of inventory management techniques for perishable goods throughout history. Goyal and Giri (2001) reviewed current developments in modelling perishable inventory, providing a comprehensive overview of the methodologies and approaches employed in this field. Their study contributed to the understanding of the challenges and opportunities associated with managing deteriorating items in various industries. Abad (2001) studied ideal pricing and order-sizing strategies for a distributor, focusing on scenarios involving partial backlog management, addressing the trade-offs between pricing decisions and inventory management strategies in a dynamic market environment. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 305 https://internationalpubls.com Ouyang and Cheng (2005) presented an inventory model designed for perishable goods with an exponential decrease in demand, incorporating the concept of partial backlog management, aiming to optimize inventory policies while considering the dynamics of demand and backlog fulfilment. Tripathy and Mishra (2010) examined inventory replenishment strategies for Weibull deteriorating items with quadratic demand and allowable payment delays, contributing to the body of literature on inventory management amidst uncertainty and credit limitations. Singh and Pattnayak (2012, 2013) proposed EOQ models for perishable goods with time-dependent demand, variable deterioration, also partial backlog, offering practical insights into inventory optimization strategies under various demand and cost structures. Amutha and Chandrasekaran (2013) investigated an economic order quantity model for perishable goods with quadratic demand and time-dependent holding charges, highlighting the importance of considering nonlinear cost functions in inventory management decisions. Dash, Singh, and Pattnayak (2014) proposed an inventory model for perishable items with exponentially decreasing demand and holding costs that vary over time, addressing the complexities of inventory management amidst fluctuating cost structures. Dutta and Kumar (2015) formulated a model for managing partially backlogged inventory of deteriorating items, considering fluctuating demand and holding costs over time, offering insights into inventory management strategies under uncertain demand and cost conditions. Uthayakumar and Karuppasamy (2017) explored a model for managing inventory of variable deteriorating pharmaceutical items, considering time-dependent demand and holding costs while utilizing trade credit, addressing the unique challenges faced by the healthcare industry in managing perishable inventory. Sekar and Uthayakumar (2018) investigated a manufacturing inventory model for coping with exponentially rising demand, incorporating preservation technology to mitigate shortages, contributing to the understanding of inventory management in dynamic production environments. Babangida and Baraya (2020) proposed a model for managing inventory of non-instantaneously perishable items with time-dependent quadratic demand, involving two storage facilities and accounting for shortages within a trade credit policy, addressing the complexities of managing inventory with multiple constraints and cost considerations. Overall, the reviewed literature underscores the importance of developing tailored inventory optimization models to address the challenges of managing deteriorating items effectively. By considering factors such as demand variability, cost structures, and backlog management strategies, businesses can enhance their inventory management practices and improve overall operational efficiency. Within this investigation, we introduce an EOQ inventory model tailored for perishable goods. Our model considers an exponentially increasing demand function and incorporates backlogging dynamics tied to the dwell time for replenishment. The main objective is to reduce total inventory expenses. We illustrate the application and solution approach of the model through numerical examples. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 306 https://internationalpubls.com Additionally, we conduct a sensitivity analysis on key parameters to evaluate their impact on the model's performance. 3. Assumptions and Notations ➒ The demand rate for the item exhibits exponential time dependence. (i.e.) 𝐷(𝑑) = π‘’πœ‡π‘‘, π‘€β„Žπ‘’π‘Ÿπ‘’ πœ‡ 𝑖𝑠 π‘π‘œπ‘›π‘ π‘‘π‘Žπ‘›π‘‘. ➒ The lead time is constant. ➒ Shortages are permissible, and during periods of stockouts, the backlogging rate varies and is contingent upon the length of the wait for the next restocking. Specifically, the backorder rate is defined as 𝐡(𝑑) = π‘’βˆ’π›Ώ(π‘‡βˆ’π‘‘), where Ξ΄ represents the backlogging parameter with a range of 0 to 1, and the parameter (T-t) denotes the dwell time, where 𝑑1 ≀ 𝑑 ≀ 𝑇. ➒ 𝐼1(𝑑) – stock level at time t, 0 ≀ 𝑑 ≀ 𝑑1. ➒ 𝐼2(𝑑) – stock level at time t, 𝑑1 ≀ 𝑑 ≀ 𝑇. ➒ T – duration of the inventory cycle. ➒ 𝑑1 – The duration of time during which the inventory remains without any shortage. ➒ CO – Ordering cost (OC). ➒ Cp – Purchase cost (PC). ➒ Ch – Holding cost (HC) is constant. ➒ πœƒ – The constant deterioration rates. ➒ Cd – The cost of deteriorated items (DC). ➒ Cs – The cost associated with shortages of backlogged items. ➒ CL – The financial impact resulting from lost sales. ➒ 𝛿 – Backlogging parameter. ➒ 𝐡(𝑑) – Backlog inventory level ➒ CT – Total cost (TC). 4. Mathematical Formulation Time T t1 Q BI Q’ Lost sales Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 307 https://internationalpubls.com Let I1(t) represent the current stock at any time instant t (0 ≀ 𝑑 ≀ 𝑑1). An stock level is Q at t = 0 and the stock level drops to zero at t = 𝑑1. The proportion of variation of inventory level is specified by, 𝑑𝐼1(𝑑) 𝑑𝑑 + πœƒπΌ1(𝑑) = βˆ’π‘’πœ‡π‘‘, 0 ≀ 𝑑 ≀ 𝑑1 through boundary conditions I1(0) = Q & I1(𝑑1)=0. The solution of (1) is 𝐼1(𝑑) = (𝑑1βˆ’π‘‘) 1+πœƒπ‘‘ [1 + (πœ‡+πœƒ) 2 (𝑑1 + 𝑑)] And 𝑄 = 𝑑1 + (πœ‡ + πœƒ) 𝑑1 2 2 During the interval [𝑑1,T], shortages occur also the demand is partially backlogged. Let 𝐼2(𝑑) represent the inventory level at 𝑑(𝑑1 ≀ 𝑑 ≀ 𝑇) then the differential equation is 𝑑𝐼2(𝑑) 𝑑𝑑 = βˆ’π‘’πœ‡π‘‘π‘’βˆ’π›Ώ(π‘‡βˆ’π‘‘), 𝑑1 ≀ 𝑑 ≀ 𝑇 With boundary conditions 𝑑 = 𝑑1, 𝐼2(𝑑) = 0. The solution of (2) is 𝐼2(𝑑) = (𝑑1 βˆ’ 𝑑) [1 βˆ’ 𝛿𝑇 + (πœ‡ + 𝛿) 2 (𝑑1 + 𝑑)] The highest level of backordered inventory denoted as BI is reached at t=T. 𝐡𝐼 = βˆ’πΌ2(𝑑) = βˆ’(𝑑1 βˆ’ 𝑇) [1 + (πœ‡ + 𝛿) 2 𝑑1 + (πœ‡ βˆ’ 𝛿) 2 𝑇] Therefore, the total order quantity during the entire time interval [0,T] is Q’=Q+BI= 𝑑1 + (πœ‡ + πœƒ) 𝑑1 2 2 βˆ’ (𝑑1 βˆ’ 𝑇) [1 + (πœ‡+𝛿) 2 𝑑1 + (πœ‡βˆ’π›Ώ) 2 𝑇] Holding charge (𝐻𝐢) = ∫ πΆβ„ŽπΌ1(𝑑)𝑑𝑑 𝑑1 0 𝐻𝐢 = πΆβ„Žπ‘‘1 2 24 [12 + 4𝑑1(πœƒ + 2πœ‡) βˆ’ 3πœƒπ‘‘1 2(πœ‡ + πœƒ)] Ordering charge (OC) = C0 Purchase charge (PC) = Cp Q’ PC = Cp [𝑑1 + (πœ‡ + πœƒ) 𝑑1 2 2 βˆ’ (𝑑1 βˆ’ 𝑇) [1 + (πœ‡+𝛿) 2 𝑑1 + (πœ‡βˆ’π›Ώ) 2 𝑇]] Deterioration charge (DC) = 𝐢𝑑 {𝑄 βˆ’ ∫ 𝐷(𝑑)𝑑𝑑 𝑑1 0 } = Cd [ πœƒπ‘‘1 2 2 ] Shortage Charge = βˆ’πΆπ‘  ∫ 𝐼2(𝑑)𝑑𝑑 𝑇 𝑑1 (1) (2) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 308 https://internationalpubls.com = 𝐢𝑠(𝑑1 βˆ’ 𝑇) [( 1βˆ’π›Ώπ‘‡ 2 ) (𝑑1 βˆ’ 𝑇) βˆ’ (πœ‡+𝛿) 6 (𝑇2 + 𝑇𝑑1 βˆ’ 2𝑑1 2)] Lost sales charge = 𝐢𝐿 ∫ [1 βˆ’ π‘’βˆ’π›Ώ(π‘‡βˆ’π‘‘)]π‘’πœ‡π‘‘π‘‘π‘‘ 𝑇 𝑑1 = 𝐢𝐿 𝛿 2 (𝑇 βˆ’ 𝑑1)2 Total cost = Ordering charge + Holding charge + Purchase charge + Deterioration charge + Shortage charge + Lost sales charge CT = C0 + πΆβ„Žπ‘‘1 2 24 [12 + 4𝑑1(πœƒ + 2πœ‡) βˆ’ 3πœƒπ‘‘1 2(πœ‡ + πœƒ)] +Cp [𝑑1 + (πœ‡ + πœƒ) 𝑑1 2 2 βˆ’ (𝑑1 βˆ’ 𝑇) [1 + (πœ‡+𝛿) 2 𝑑1 + (πœ‡βˆ’π›Ώ) 2 𝑇]]+ Cd [ πœƒπ‘‘1 2 2 ] +𝐢𝑠(𝑑1 βˆ’ 𝑇) [( 1βˆ’π›Ώπ‘‡ 2 ) (𝑑1 βˆ’ 𝑇) βˆ’ (πœ‡+𝛿) 6 (𝑇2 + 𝑇𝑑1 βˆ’ 2𝑑1 2)] + 𝐢𝐿 𝛿 2 (𝑇 βˆ’ 𝑑1)2 Our goal is to minimize the overall cost. The necessary condition is πœ•πΆπ‘‡ πœ•π‘‘1 = 0 π‘Žπ‘›π‘‘ πœ•2𝐢𝑇 πœ•π‘‘1 2 > 0 π‘“π‘œπ‘Ÿ π‘Žπ‘™π‘™ 𝑑1 > 0 We get πœ•πΆπ‘‡ πœ•π‘‘1 = πΆβ„Ž [𝑑1 + πœƒπ‘‘1 2 2 + πœ‡π‘‘1 2 βˆ’ πœ‡πœƒπ‘‘1 3 2 βˆ’ πœƒ2𝑑1 3 2 ] + 𝐢𝑝[πœƒπ‘‘1 βˆ’ 𝛿𝑑1 + 𝛿𝑇] + πΆπ‘‘πœƒπ‘‘1 + 𝐢𝐿[βˆ’π›Ώπ‘‡ + 𝛿𝑑1] + 𝐢𝑠 [𝑑1 βˆ’ 𝑇 βˆ’ 𝛿𝑇𝑑1 + 𝛿𝑇2 βˆ’ πœ‡π‘‡ 6 + 2 3 𝑑1πœ‡ βˆ’ 𝛿𝑇 6 + 2 3 𝛿𝑑1] = 0 And πœ•2𝐢𝑇 πœ•π‘‘1 2 = πΆβ„Ž [1 + πœƒπ‘‘1 + 2πœ‡π‘‘1 βˆ’ 3 2 πœƒπ‘‘1 2(πœ‡ + πœƒ)] + 𝐢𝑝[πœƒ βˆ’ 𝛿] + πΆπ‘‘πœƒ + 𝐢𝐿𝛿 + 𝐢𝑠 [1 βˆ’ 𝛿𝑇 + 2 3 (πœ‡ + 𝛿)] > 0 5. Numerical illustration 1: Let's examine an inventory system characterized by the following parameter values, expressed in appropriate units [πœƒ, 𝑇, πœ‡, 𝛿, πΆπ‘œ , 𝐢𝑑, πΆβ„Ž, 𝐢𝑝, 𝐢𝑠, 𝐢𝐿] = [0.03,5,1.5,0.001,2,0.5,0.1,30,3,1]. Then we get t1 =2.8203, Q’=23.7596 and CT=761.3449 Numerical illustration 2: Let's examine an inventory system characterized by the following parameter values, expressed in appropriate units [πœƒ, 𝑇, πœ‡, 𝛿, πΆπ‘œ , 𝐢𝑑, πΆβ„Ž, 𝐢𝑝, 𝐢𝑠, 𝐢𝐿] = [0.1,4,2.5,0.005,3,0.3,1,30,4,1.5]. Then we get t1 = 1.2475, Q’=24.0589 and CT = 824.1926 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 309 https://internationalpubls.com 6. Sensitivity Analysis Table – 1 : Fluctuation in deterioration rate (πœƒ) 𝜽 t1 Q’ CT 0.01 2.8447 23.7517 760.3105 0.02 2.8324 23.7557 760.8297 0.03 2.8203 23.7596 761.3449 0.04 2.8082 23.7634 761.8560 0.05 2.7962 23.7671 762.3629 Table-1 shows that when deterioration rate πœƒ increases, automatically quantity, total cost are increases and the duration of time during which the inventory remains without any shortage (t1) is decreases. Table – 2 : Fluctuation in length of the inventory cycle (T) T t1 Q’ CT 4.6 2.6076 20.4782 652.7475 4.8 2.7142 22.0889 705.9319 5.0 2.8203 23.7596 761.3449 5.2 2.9259 25.4903 819.0066 5.4 3.0310 27.2810 878.9372 Table-2 shows that when T increases, automatically the duration of time during which the inventory remains without any shortage (t1), quantity and total cost are increases. Table – 3 : Fluctuation in Β΅ Β΅ t1 Q’ CT 1.3 2.9414 21.2609 677.6937 1.4 2.8786 22.5102 719.4678 1.5 2.8203 23.7596 761.3449 1.6 2.7659 25.0090 803.3134 1.7 2.7152 26.2584 845.3630 Table-3 shows that when Β΅ increases, automatically quantity, total cost are increases and the duration of time during which the inventory remains without any shortage (t1) is decreases. 7. Conclusion In conclusion, the development of an inventory optimization model tailored specifically for perishable goods with exponential demand variation, stable holding charges, also partial backlog management represents a notable advancement in the realm of supply chain management. By addressing the complexities inherent in managing such inventory items, this model offers valuable insights and practical guidance for businesses striving to achieve efficient inventory management practices. Through numerical examples and sensitivity analyses, the effectiveness and robustness of the proposed model have been demonstrated, highlighting its potential to enhance operational Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 310 https://internationalpubls.com efficiency, minimize costs, and improve customer satisfaction. As businesses continue to face evolving market dynamics and supply chain challenges, the implementation of advanced inventory optimization models like the one presented here can serve as a strategic tool for driving sustainable growth and competitive advantage. Future extensions of the inventory optimization model could include advanced demand forecasting techniques, integration of dynamic pricing strategies, implementation of real-time inventory tracking using IoT technology, and consideration of sustainability metrics for environmentally conscious inventory management. References [1] Su C.T., Lin C.W., and Tsai C.H. (1999): β€œA deterministic production inventory model for deteriorating items with an exponential declining demand”, Operational Research Society of India, 36 (2), 95-106. [2] N. H. Shah and Y. K. Shah (2000): β€œLiterature Survey on Inventory Model For Deteriorating Items”, Economics Annals, Vol. 44, pp. 221-237. [3] S. K. Goyal and B. C. 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(2017): β€œAn inventory model for variable deteriorating pharmaceutical items with time dependent demand and time dependent holding cost under trade credit in healthcare industries”, Commun. Appl. Anal. 210(4), 533–549. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 3s (2024) 311 https://internationalpubls.com [17] Sekar T., Uthayakumar R. (2018): β€œA manufacturing inventory model for exponentially increasing demand with preservation technology and shortage”, Int. J. Oper. Res. 15(2), 61–70. [18] Babangida B., Baraya Y.M. (2020): β€œAn inventory model for non-instantaneous deteriorating items with time dependent quadratic demand, two storage facilities and shortages under trade credit policy”, Int. J. Model. Oper. Manag. 8(1), 1–44.