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A Novel EOQ Model for Deteriorating Items with Non-Zero Initial Ramp-Type Demand and Two-Phase Discounting

This paper proposes a novel EOQ model for deteriorating items that incorporates a non-zero initial ramp-type demand and a two-phase discounting policy to optimize inventory management under time-dependent deterioration and no shortages.

Original authors: Rina Khatun, Shariful Alam, Khairul Habib Alam

Published 2026-09-16

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Rina Khatun, Shariful Alam, Khairul Habib Alam

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a world where every product you buy has a ticking clock, not just for when it expires, but for how its value and appeal change the moment it sits on a shelf. This is the reality for industries dealing with perishable goods, from fresh produce and medicine to fashion and electronics. These items do not sit still; they slowly lose quality over time, a process known as deterioration. At the same time, the desire for these items is rarely constant. It often starts with a burst of interest, perhaps sparked by a new advertisement or a pre-order campaign, grows as more people hear about it, and eventually settles into a steady rhythm once the market is saturated. For decades, the mathematical tools used to manage these goods have been too simple, often assuming that demand starts at zero and stays flat, or that items only begin to rot after a long, static period. These simplifications leave business managers without a clear map for navigating the complex, shifting landscape of real-world inventory.

A new study by researchers at IIEST Shibpur and SRM University seeks to correct this by building a more realistic model for how to stock and sell deteriorating items. Instead of relying on rigid, outdated assumptions, the authors constructed a system that accounts for the fact that demand often begins at a positive level due to marketing efforts, then rises before stabilizing. They also recognized that the way a product degrades is often linked directly to how long it has been stored, with the rate of decline increasing over time. To make the model even more practical, the researchers introduced a strategy of using different discount prices before the item starts to deteriorate and again after it begins to lose quality. This two-phase approach allows businesses to stimulate early sales and then adjust prices to clear stock before it becomes unsellable, all while ensuring that the store never runs out of product.

The researchers developed a mathematical framework to calculate the perfect moment to reorder stock and the ideal amount to order, aiming to maximize profit over an endless timeline. They tested this model using a specific set of conditions that mimic a real business scenario. In their simulation, they set the initial demand to a specific positive number, representing the immediate interest generated by advertising, and let it grow linearly before transitioning into a steady, stable demand pattern. They applied a linear deterioration rate, meaning the items lose value faster as time passes, and used distinct discount factors for the periods before and after this deterioration began. By running these calculations on a computer, they found a specific cycle time of approximately 4.40 units of time that yielded the highest average profit, which came out to roughly 33,240 units of currency, with an optimal initial inventory level of about 6,913 items.

To ensure their findings were robust and not just a fluke of one specific set of numbers, the team performed a series of sensitivity tests. They systematically changed one key factor at a time while keeping everything else constant to see how the system reacted. When they increased the initial demand driven by advertising, they observed that the total profit rose significantly, confirming that early marketing efforts are a powerful lever for financial success. However, when they increased the rate at which the items deteriorated, the results were starkly different. A faster decay rate forced the business to shorten its ordering cycles, reduce the amount of stock held, and accept a sharp drop in overall profit. This highlighted that managing the speed of deterioration is critical; even small increases in how quickly items spoil can erode financial performance.

The study also explored how different pricing strategies and costs influenced the outcome. They found that the selling price of the item had a strong, positive effect on profit, while the cost to purchase the items and the cost to hold them in storage acted as brakes, reducing both the optimal inventory level and the total earnings. Interestingly, the specific shape of the demand curve and the exact timing of when deterioration started had a mixed effect, sometimes extending the ordering cycle and sometimes shortening it, depending on the balance between how fast the market wanted the product and how fast the product was spoiling. The researchers concluded that while their model is a significant step forward in understanding these dynamics, it is a simulation based on specific assumptions, such as the absence of stock shortages. They suggest that future work could expand this model to include situations where running out of stock is possible or to account for more complex supply chain realities, but for now, this approach offers a clearer, more adaptable tool for managers trying to balance the ticking clock of product decay with the rising tide of customer demand.

Technical Summary: A Novel EOQ Model for Deteriorating Items with Non-Zero Initial Ramp-Type Demand and Two-Phase Discounting

Problem Statement
The paper addresses the challenge of managing inventory for deteriorating items under realistic market conditions that classical Economic Order Quantity (EOQ) models often fail to capture. Traditional models typically assume constant demand, zero initial demand, and ignore the dynamic interplay between advertising, product life cycles, and time-dependent deterioration. The authors identify a gap in existing literature where factors such as advertising-driven non-zero initial demand, general ramp-type demand stabilization, time-proportional deterioration, and distinct pre- and post-deterioration discounting policies are rarely integrated into a single, unified framework. The objective is to develop an infinite-horizon, no-shortage inventory model that simultaneously accounts for these complex, real-world variables to optimize replenishment cycles and maximize average profit.

Methodology
The study formulates a deterministic inventory model for a single-item system with an infinite planning horizon and no shortages. The methodology involves the following key components:

Demand Structure:
- Pre-deterioration Phase ( $0 \le t \le d$ ): Demand is modeled as a linear function $D_1(t) = \alpha + \beta t$ , where $\alpha > 0$ represents a non-zero initial demand driven by advertising or pre-orders, and $\beta$ represents the growth rate.
- Post-deterioration Phase ( $t \ge d$ ): Demand follows a general ramp-type function $D_2(t)$ that increases until a stabilization time $r$ , after which it remains constant at $f(r)$ .
- Discount Adjustment: The model incorporates a two-phase discounting scheme. Demand in the pre-deterioration phase is scaled by a factor $d_a = (1-d_1)^{-p}$ , and in the post-deterioration phase by $d_b = (1-d_2)^{-q}$ , where $d_1$ and $d_2$ are discount rates and $p, q$ control their intensity.
Deterioration:
- Deterioration is assumed to be time-dependent and proportional to time, represented by the rate $D(t) = \xi t$ . This linear form reflects a gradual loss of quality or utility during storage.
Mathematical Formulation:
- The inventory levels are governed by a system of ordinary differential equations (ODEs) across three intervals: before deterioration ( $0 \le t \le d$ ), during the ramp-up of demand after deterioration ( $d \le t \le r$ ), and during the stabilization phase ( $r \le t \le t_1$ ).
- The authors derive closed-form solutions for the inventory level $I(t)$ and the total units demanded $Q(t)$ by integrating these ODEs with specific boundary conditions ( $I(0)=I$ and $I(t_1)=0$ ).
- An average profit function ( $AP$ ) is constructed, incorporating revenue, purchasing costs, holding costs, deterioration costs, ordering costs, and advertisement costs.
Optimization:
- The optimal cycle time $t_1^*$ is determined by solving the first-order condition $\frac{d(AP)}{dt_1} = 0$ . This leads to a non-linear equation involving the derivative of the total profit with respect to time (Eq. 25).
- Closed-form formulas for the optimal inventory policy are constructed, and numerical illustrations are provided using MATLAB to demonstrate the application of these formulas and perform sensitivity analysis.

Key Results
The paper presents a numerical example and a comprehensive sensitivity analysis to validate the model and explore parameter impacts:

Optimal Solution: Using a specific set of parameters (e.g., $\alpha=53, \beta=3, \xi=0.073$ ), the model yields an optimal cycle time $t_1^* \approx 4.3985$ , an initial inventory level $I^* \approx 6913.18$ , and an average profit $AP^* \approx 33,240.42$ .
Sensitivity Analysis:
- Deterioration Rate ( $\xi$ ): This is identified as a critical factor. As $\xi$ increases, the optimal cycle time shortens, inventory levels drop sharply, and average profit decreases significantly. This highlights the financial penalty of rapid deterioration.
- Demand Parameters ( $\alpha, \beta$ ): Increases in initial demand ( $\alpha$ ) and growth rate ( $\beta$ ) lead to higher inventory levels and significantly improved profitability, though the cycle time remains relatively stable.
- Discounting ( $d_1, d_2$ ): The post-deterioration discount ( $d_2$ ) has a more constructive impact on profit than the pre-deterioration discount ( $d_1$ ).
- Costs ( $C, h$ ): Increases in purchasing cost ( $C$ ) and holding cost ( $h$ ) negatively impact both inventory levels and profitability, necessitating shorter cycle times to mitigate losses.
- Stabilization Time ( $r$ ): A longer stabilization period generally supports higher profitability and inventory levels.

Significance and Contributions
The paper claims to bridge specific research gaps by integrating four distinct characteristics into a single EOQ framework:

Non-Zero Initial Demand: Unlike classical ramp-type models that start at zero, this model accounts for initial demand generated by advertising, offering a more realistic representation of product launches.
General Ramp-Type Demand: It utilizes a flexible demand function that stabilizes after a specific time, capturing the full product life cycle from growth to maturity.
Two-Phase Discounting: The model introduces distinct discount policies for pre- and post-deterioration phases, providing a mechanism to manage demand and losses differently as the product ages.
Unified Infinite-Horizon Framework: It combines advertising effects, time-proportional deterioration, and dual discounts in a no-shortage, infinite-horizon setting, extending existing models that typically treat these factors in isolation.

The authors conclude that the proposed model offers managers an advanced tool for pricing and replenishment decisions, particularly for industries dealing with perishable or degradable goods (e.g., food, medicine, electronics) where demand dynamics and deterioration rates are critical. The study suggests that controlling deterioration and optimizing demand growth through advertising and strategic discounting are key levers for enhancing system stability and profitability. Future work is suggested to extend the model to include shortages, trade credit, or stochastic demand.

Technical Summary: A Novel EOQ Model for Deteriorating Items with Non-Zero Initial Ramp-Type Demand and Two-Phase Discounting

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