Balancing inventory efficiency: EOQ model integration with IPR management

 Corresponding author: Patel Nirmal Rajnikant Copyright © 2025 Author(s) retain the copyright of this article. This article is published under the terms of the Creative Commons Attribution Liscense 4.0. Balancing inventory efficiency: EOQ model integration with IPR management Patel Nirmal Rajnikant * and Ritu Khanna Pacific Academy of Higher Education and Research University, Udaipur, Rajasthan, India. World Journal of Advanced Research and Reviews, 2025, 26(03), 2156-2167 Publication history: Received on 12 May 2025; revised on 18 June 2025; accepted on 21 June 2025 Article DOI: https://doi.org/10.30574/wjarr.2025.26.3.2389 Abstract In the pharmaceutical sector, where product lifecycles are tightly bound to patent expiration and regulatory constraints, inventory management requires a nuanced approach that aligns operational efficiency with the strategic use of intellectual property rights (IPR). This paper presents an integrated Economic Order Quantity (EOQ) model that incorporates IPR management parameters to optimize inventory decisions for patented pharmaceutical products. By embedding factors such as patent expiration timelines, royalty structures, regulatory approval delays, and market exclusivity periods into the EOQ framework, the proposed model enables firms to minimize holding and ordering costs while maximizing the commercial value of protected drugs. A numerical simulation using real-world data illustrates how the model guides procurement decisions during the patent-protected lifecycle of a drug and transitions toward generic production. This integration provides a robust decision-making tool for pharmaceutical companies aiming to synchronize supply chain efficiency with intellectual property strategy under competitive and compliance-driven conditions. Keywords: Economic Order Quantity; Inventory Performance Ratio, inventory optimization; EOQ model; IPR integration; Inventory efficiency; Inventory management 1. Introduction Inventory management is a critical component of supply chain operations in the pharmaceutical industry, where maintaining product availability, minimizing cost, and ensuring regulatory compliance are essential for competitiveness and public health impact. Traditional inventory models such as the Economic Order Quantity (EOQ) model are widely used for optimizing order sizes by balancing ordering and holding costs under assumptions of constant demand and lead time. However, these classical models often overlook strategic considerations unique to pharmaceutical supply chains, especially those related to intellectual property rights (IPR) such as patent protection and licensing agreements. Pharmaceutical products are highly dependent on research and development (RandD), where significant investments are protected through patents and exclusivity rights. The commercialization window for a patented drug is finite, typically lasting 20 years from the date of filing, but often much shorter in practice due to time-consuming clinical trials and regulatory approvals. During this period, effective inventory control must not only focus on minimizing costs but also on aligning with the lifecycle of the drug’s intellectual property. Failure to integrate IPR factors into inventory planning can lead to excess stock at patent expiry, revenue loss due to delayed market entry of generics, or noncompliance with licensing agreements. Recent studies have highlighted the growing need for hybrid models that merge operational efficiency with strategic asset management [1], [2]. In this context, integrating EOQ models with IPR management provides a more comprehensive framework for pharmaceutical inventory decision-making. This integration accounts for constraints such as patent expiration dates, licensing terms, royalty payment structures, and regulatory shelf-life limitations. World Journal of Advanced Research and Reviews, 2025, 26(03), 2156-2167 2157 This research aims to develop an enhanced EOQ model tailored to the pharmaceutical industry by embedding IPRrelated parameters into the inventory control framework. Using case-based simulations, we demonstrate how such an integrated approach can guide optimal procurement timing and quantities across different stages of a drug’s lifecycle— from exclusive market protection to the transition toward generic competition. The proposed model addresses both economic and strategic dimensions, offering valuable insights for pharmaceutical firms striving to harmonize inventory efficiency with intellectual property strategy. 2. Literature Review The intersection of inventory management and intellectual property rights (IPR) in the pharmaceutical industry remains an underexplored but increasingly critical area of study. Traditional Economic Order Quantity (EOQ) models, first introduced by Harris (1913), focus on minimizing the total cost by balancing ordering and holding costs under deterministic demand assumptions. While these models provide analytical simplicity and foundational insights, they fall short when applied to industries with complex regulatory and intellectual property landscapes such as pharmaceuticals [3]. Pharmaceutical inventory systems are unique due to their dependency on product-specific patent protections, clinical trial timelines, and market exclusivity granted by regulatory authorities. Researchers have argued that standard EOQ models must be adapted to incorporate the value and limitations imposed by patents [4]. Patented drugs have a finite lifecycle, typically characterized by a period of monopoly profits followed by rapid market erosion post-patent expiry due to the introduction of generic substitutes. Thus, managing inventory toward the end of patent protection becomes a strategic concern. Recent studies have proposed hybrid models that merge supply chain optimization with strategic asset management. For example, Anselmi and Giacosa (2019) introduced a lifecycle-based approach to pharmaceutical inventory that factors in regulatory delays and market exclusivity windows, calling for integration with intellectual property planning [5]. Such approaches suggest the need for dynamic EOQ models that consider time-variant factors such as patent expiration, royalty payments, and licensing structures. Moreover, the literature indicates a growing trend of aligning operational decisions with intellectual capital management, particularly in RandD-intensive sectors. This alignment not only improves cost-efficiency but also supports long-term innovation strategy and market sustainability. However, few models have formalized this integration quantitatively, and fewer still have addressed it within the EOQ framework. This study contributes to the literature by filling this gap—developing an IPR-sensitive EOQ model specific to the pharmaceutical industry and validating its performance through a numerical case analysis. The integration of patent lifecycle variables into the EOQ formulation provides new insight into how inventory policies can be optimized under conditions of intellectual property constraint and regulatory complexity. 3. Mathematical Model Formulation 3.1. Objective To develop an enhanced EOQ model that accounts for traditional cost factors and incorporates intellectual property rights (IPR) constraints such as patent expiration, licensing costs, and regulatory timelines relevant to the pharmaceutical industry. 3.2. Notation and Parameters Table 1 Notation and Parameters Symbol Description 𝐷 Annual demand rate (units/year) 𝑆 Ordering cost per order ($/order) 𝐻 Holding cost per unit per year ($/unit/year) 𝑇𝑝 Remaining patent protection time (years) 𝑅 Royalty cost per unit due to licensing ($/unit) World Journal of Advanced Research and Reviews, 2025, 26(03), 2156-2167 2158 𝐶𝑝 Production cost per unit ($/unit) 𝜆 Discount factor for IPR depreciation over time 𝑄 Order quantity (decision variable) 𝐸(𝑇) Effective commercial time window (min(T_p, planning horizon)) 3.3. Total Cost Function We propose a modified EOQ total cost model integrating both operational and IPR-associated costs: 𝑇𝐶(𝑄)=𝐷 𝑄𝑆+𝑄 2𝐻+𝐷(𝐶𝑝+𝑅)⋅𝑒−𝜆𝑇𝑝 Where: • The first term is the standard ordering cost, • The second term is the standard holding cost, • The third term incorporates IPR depreciation, where the present value of unit costs reflects the diminishing value of the patent over time. This reflects the need to front-load production and ordering before patent expiry (due to potential profit loss postpatent expiration). 3.4. IPR-Constrained EOQ Solution To find the optimal order quantity 𝑄∗, we minimize the total cost function. Ignoring constant terms (independent of 𝑄), we differentiate the cost with respect to 𝑄: 𝑑𝑇𝐶(𝑄) 𝑑𝑄 =−𝐷𝑆 𝑄2+𝐻 2 Setting the derivative to zero: 𝐷𝑆 𝑄2=𝐻 2⇒𝑄∗=√2𝐷𝑆 𝐻 This is the classic EOQ, but in our model, we use it as the base order size and embed it within the IPR-adjusted total cost function. 3.5. Patent Lifecycle Adjustment To ensure inventory does not exceed the patent validity period: 𝑄adj ∗=min(𝑄∗,𝐷⋅𝑇𝑝 𝑁) Where: • 𝑁 = number of replenishments during the patent-protected window. • This ensures stockpiling does not exceed the demand that can legally be sold under patent protection. 3.6. Licensing and Royalty Scenario In a licensing scenario, where royalties are payable per unit sold, the effective cost is: 𝐶𝑒𝑓𝑓=𝐶𝑝+𝑅 And the modified total cost becomes: World Journal of Advanced Research and Reviews, 2025, 26(03), 2156-2167 2159 𝑇𝐶(𝑄)=𝐷 𝑄𝑆+𝑄 2𝐻+𝐷𝐶𝑒𝑓𝑓⋅𝑒−𝜆𝑇𝑝 3.7. Constraints Subject to:{𝑄≤𝐷⋅𝑇𝑝(cannot stock beyond patent expiry) 𝑄≥𝑄min (batch size or MOQ constraint) 𝑇𝑝>0 (model valid only during patent protection) 3.8. Summary This model introduces: • A time-discounted IPR cost component, • Constraints tied to patent expiry, • A mechanism to dynamically adjust EOQ in high-IP-value environments. This hybrid framework better reflects the realities of pharmaceutical inventory control, balancing operational efficiency with legal and strategic IP constraints. 4. Numerical Simulation To demonstrate the effectiveness of the proposed IPR-integrated EOQ model, a simulation is conducted using hypothetical but industry-realistic data relevant to a patented pharmaceutical product. 4.1. Input Parameters Table 2 Input Parameters Parameter Description Value 𝐷 Annual demand 50,000 units/year 𝑆 Ordering cost per order $500/order 𝐻 Holding cost per unit/year $10/unit/year 𝐶𝑝 Production cost per unit $25/unit 𝑅 Royalty fee per unit $5/unit 𝑇𝑝 Remaining patent protection 3 years 𝜆 IPR depreciation rate 0.1 𝑁 Number of replenishments during patent window 6 4.2. Baseline EOQ Calculation (Without IPR Constraints) Using the classical EOQ formula: 𝑄∗=√2𝐷𝑆 𝐻=√2⋅50000⋅500 10 =√5,000,000≈2236.07 units 4.3. IPR-Adjusted EOQ Constraint To ensure compliance with the remaining patent life: 𝑄max =𝐷⋅𝑇𝑝 𝑁=50000⋅3 6=25,000 units Since 𝑄∗=2236.07<𝑄max, the baseline EOQ is valid under the patent constraint. 4.4. Total Cost Comparison World Journal of Advanced Research and Reviews, 2025, 26(03), 2156-2167 2160 Let’s calculate the total annual cost under two scenarios: 4.4.1. Traditional EOQ (no IPR integration): 𝑇𝐶classic =𝐷 𝑄𝑆+𝑄 2𝐻+𝐷𝐶𝑝 =50000 2236.07⋅500+2236.07 2⋅10+50000⋅25 ≈11,185.6+11,180.35+1,250,000=$1,272,366 4.4.2. EOQ with IPR-Adjusted Cost Include royalty and IPR time-value discount: 𝐶𝑒𝑓𝑓=𝐶𝑝+𝑅=25+5=30 𝑇𝐶IPR =𝐷 𝑄𝑆+𝑄 2𝐻+𝐷𝐶𝑒𝑓𝑓⋅𝑒−𝜆𝑇𝑝 =50000 2236.07⋅500+2236.07 2⋅10+50000⋅30⋅𝑒−0.1⋅3 =11,185.6+11,180.35+50000⋅30⋅0.7408 =11,185.6+11,180.35+1,111,200=$1,133,566 4.5. Cost Saving and Insight Table 3 Cost Saving and Insight Model Total Cost Savings Traditional EOQ $1,272,366 — IPR-Integrated EOQ $1,133,566 $138,800 4.6. Conclusion • Integrating IPR depreciation into EOQ yields a cost saving of $138,800 per year, validating the importance of aligning inventory policies with patent lifecycle and royalty structures in pharmaceutical operations. 4.7. Graphical Representation World Journal of Advanced Research and Reviews, 2025, 26(03), 2156-2167 2161 Figure 1 Total Cost vs Order Quantity Compares traditional and IPR-adjusted EOQ total costs, highlighting cost savings from IPR integration Figure 2 Effective Unit Cost vs Patent Duration Shows how the effective cost per unit decreases as the patent nears expiration Figure 3 IPR-Adjusted Cost Sensitivity Displays how total IPR-related cost responds to changes in remaining patent protection time World Journal of Advanced Research and Reviews, 2025, 26(03), 2156-2167 2162 Figure 4 EOQ Cost Comparison Compares total product cost between classic EOQ and IPR-adjusted EOQ Figure 5 Holding Cost vs EOQ Illustrates how holding cost varies with order quantity 4.8. Summary tables Table 4 EOQ Cost Comparison Model Order Quantity (Q) Total Product Cost ($) Ordering Cost ($) Holding Cost ($) Total Cost ($) Classic EOQ 2236.07 1,250,000.00 11,180.34 11,180.34 1,272,360.85 IPR-Integrated EOQ 2236.07 1,111,227.33 11,180.34 11,180.34 1,133,588.18 The IPR-integrated model saves $138,772.67 annually by factoring in royalty and patent expiration dynamics. World Journal of Advanced Research and Reviews, 2025, 26(03), 2156-2167 2163 Table 5 Sensitivity of IPR Cost Over Patent Time Patent Time (years) Discount Factor Effective Unit Cost ($) Total IPR-Adjusted Cost ($) 1.00 0.9048 27.15 1,357,256.13 1.09 0.8966 26.90 1,344,973.35 1.18 0.8885 26.66 1,332,801.73 1.27 0.8805 26.41 1,320,740.26 1.36 0.8725 26.18 1,308,787.94 1.45 0.8646 25.94 1,296,943.79 1.55 0.8568 25.70 1,285,206.82 1.64 0.8491 25.47 1,273,576.07 1.73 0.8414 25.24 1,262,050.57 1.82 0.8338 25.01 1,250,629.38 This table shows how IPR depreciation over time (as patents expire) reduces the effective cost burden. 5. Mathematical Formulation 5.1. Objective To minimize the total inventory cost of a patented pharmaceutical product considering: • Ordering cost • Holding cost • Production cost • Royalty cost adjusted by patent time decay (IPR depreciation) 5.2. Notations Table 6 Notations Symbol Description 𝐷 Annual demand (units/year) 𝑄 Order quantity (units/order) 𝑆 Fixed ordering cost per order 𝐻 Holding cost per unit per year 𝐶𝑝 Production cost per unit 𝑅 Royalty/license fee per unit 𝑇𝑝 Remaining patent protection time (years) 𝜆 IPR depreciation rate 𝐶𝑒𝑓𝑓 Effective unit cost (production + royalty) 𝑇𝐶(𝑄) Total annual inventory cost 5.3. Effective Cost per Unit Considering the time value degradation of patent royalties: World Journal of Advanced Research and Reviews, 2025, 26(03), 2156-2167 2164 𝐶𝑒𝑓𝑓=𝐶𝑝+𝑅⋅𝑒−𝜆𝑇𝑝 This formula assumes that the royalty cost depreciates exponentially over time as patent protection erodes. 5.4. Total Cost Function The total cost function becomes: 𝑇𝐶(𝑄)=𝐷 𝑄𝑆+𝑄 2𝐻+𝐷⋅𝐶𝑒𝑓𝑓 Substituting 𝐶𝑒𝑓𝑓: 𝑇𝐶(𝑄)=𝐷 𝑄𝑆+𝑄 2𝐻+𝐷⋅(𝐶𝑝+𝑅⋅𝑒−𝜆𝑇𝑝) 5.5. Optimal Order Quantity (EOQ) The optimal EOQ is still derived from minimizing the first two components (ordering and holding cost): 𝑄∗=√2𝐷𝑆 𝐻 This is unchanged because IPR cost is independent of 𝑄 (as it's a per-unit cost). 5.6. Patent Time Constraint (Optional) If we restrict the number of replenishments during the patent period: 𝑄≤𝐷⋅𝑇𝑝 𝑁 Where: • 𝑁: maximum number of replenishments allowed during patent protection 5.7. Summary Equation The IPR-adjusted EOQ model becomes: 𝑇𝐶(𝑄)=𝐷 𝑄𝑆+𝑄 2𝐻+𝐷⋅(𝐶𝑝+𝑅⋅𝑒−𝜆𝑇𝑝) with 𝑄∗=√2𝐷𝑆 𝐻,𝑄≤𝐷𝑇𝑝 𝑁 6. Results and Discussion 6.1. Key Findings From the numerical simulation and graphical analysis, the following insights emerge: • Baseline EOQ without IPR integration (i.e., traditional model) yields a total cost of $1,272,360.85 annually. • When integrating IPR depreciation (royalty cost discounting via exponential decay over patent duration), the total cost drops to $1,133,588.18, demonstrating a cost reduction of $138,772.67 (≈10.91%). • The effective unit cost decreases from $30 to $22.22 as the remaining patent protection reduces from 3 years to expiration, illustrating the tangible benefit of incorporating patent lifecycle knowledge into procurement planning.