An Inventory Optimization Model for Complex Systems with Imprecise Demand and Variable Holding Cost using AI Forecasting

Vinay Kumar Masiyare¹*, Animesh Kumar Sharma²

¹ Research Scholar, Department of Mathematics, The ICFAI University, Raipur, Chhattisgarh, India
² Assistant Professor, Department of Mathematics, The ICFAI University, Raipur, Chhattisgarh, India

Abstract

Handling inventory in complex systems needs holistic approach that can significantly solve the uncertainties, the deterioration and fluctuation in the holding costs. Conventional models of EOQ are based on deterministic assumptions of a fixed holding cost and demand which is always constant hence not suitable in real life situations. In order to avoid these shortcomings, this paper introduces an integrated inventory model that integrates AI assisted demand forecasting, fuzzy logic based uncertainty modelling, and deterioration related holding cost structure into a single optimization framework. The analysis of historical demand data is performed with the help of Long Short-Term Memory (LSTM) neural network, which displays nonlinear changes that time may bring and provides a highly precise forecast of 110.68 units in the next cycle. As AI predictions are fundamentally uncertain, an asymmetric triangular fuzzy number is used to reflect pessimistic, most likely, and optimistic demand cases. The fuzzy representation is subsequently defuzzified giving rise to an effective demand of 111.60 units which becomes a strong and modified input to the optimization model. The deterioration associated with holding cost has been included in the proposed model in order to ensure that the rise in storage costs attributed to perishable products is reflected realistically. The objective of the optimization is the minimization of the Average Total Cost (ATC) by establishing the optimal order quantity and cycle time. The numerical example demonstrates that the best cycle time is about 0.65, order quantity is about 72.24 units, and the lowest cost is 152.80 per cycle time. Sensitivity analysis shows that the system behaviour remains stable with changes in the demand, holding cost, deterioration cost and deterioration rate. In general, the combined method based on AI, Fuzzy and Deterioration presents a realistic, strong, and managerial applicable framework of the optimal use of inventory decisions in uncertain and dynamic conditions.

Keywords: Inventory optimization, Fuzzy demand, AI-based forecasting, LSTM, Deterioration.

1. Introduction

Production and distribution networks are based on the inventory systems. Classical models of the economic order quantity (EOQ) presuppose that the demand is deterministic and that the holding cost is constant, which is hardly realistic in the current markets of volatility. The development of complex systems that include interacting, adaptive, and uncertain systems requires an integrative modelling method that integrates artificial intelligence (AI), fuzzy logic, and classical mathematical optimization. Deterioration and uncertainty have been identified as a crucial force behind cost fluctuation in the last ten years (Karimi, 2025). Perishable or degradable products, such as food to pharmaceuticals, have a decreasing value with time and their demand is affected by various factors that are uncertain like seasonality and consumer behaviour. Additionally, the holding cost does not remain the same and in many cases, it rises with deterioration rate because of the necessity to have temperature control, maintenance, or ensure quality (Verma et al., 2024). To overcome these issues, the current studies focus on hybrid intelligent systems in which AI algorithms like long short-term memory (LSTM) networks forecast demand trends and fuzzy reasoning processes represent linguistic or imprecise variables (Tiwari et al, 2025). Integrating these tools with inventory cost optimization leads to more robust decision-making. However, few studies combine all three -AI forecasting, fuzzy demand representation, and deterioration-dependent cost structures into a single mathematical model. This paper bridges that gap by formulating a comprehensive cost-minimization model for deteriorating items in complex systems. The proposed model accounts for fuzzy demand estimated through AI prediction and derives an analytical solution using nonlinear optimization. Sensitivity analysis further validates model stability under varying system parameters.

2. Literature Review

2.1 Inventory Models under Deterioration

Early work by Singh (2018) extended EOQ models to include deterioration, assuming an exponential decay rate. Later, Shanthi & Karthikeyan (2021) analysed two-warehouse systems with time-dependent deterioration. More recent formulations link deterioration directly to holding cost: as deterioration increases, effective storage cost rises non-linearly (Vahdani et al, 2022). These studies highlight the importance of coupling cost structure with physical degradation but do not incorporate demand uncertainty.

2.2 Fuzzy Demand and Uncertainty Modelling

Fuzzy logic provides a natural mechanism for representing imprecise demand information. Li and Soni & Suthar (2021) proposed a triangular fuzzy EOQ model to handle vagueness in customer orders. Das (2022) extended this by using trapezoidal membership functions for multi-item inventory. Tiwari et al. (2024) integrated fuzzy demand with inflation and trade credit, demonstrating reduced total cost under linguistic uncertainty. However, these works rely on static fuzzy parameters and lack AI-based dynamic forecasting.

2.3 AI-Based Forecasting in Inventory Systems

Demand prediction has been revolutionized using artificial intelligence methods. LSTM and recurrent neural networks (RNNs) are more effective at modelling temporal dependencies compared to the conventional regression models (Su et al., 2024). The work by Das et al. (2024) used LSTM to forecast the demand of perishable food, which was more accurate than ARIMA. Kumar and Goyal (2025) incorporated the LSTM output in a replenishment model and optimized the stock levels dynamically. However, not many studies use a combination of those AI-based predictions with fuzzy defuzzification and cost optimization at the same time.

2.4 Research Gap and Motivation

Based on the literature that has been surveyed, it is clear that there is a research gap in the field of integrated modelling of uncertainty, deterioration and cost optimization. Although a number of studies have discussed the fuzzy demand or cost dependent on deterioration separately, a unified model of both factors in an AI-based prediction setting is still uncommon. The models that are in place assume either a fixed fuzzy parameter or a deterministic demand projection that cannot be used to reflect the real-time fluctuation associated with complex inventory set ups. In addition, the existence of dynamic relationship between demand uncertainty, deterioration and sensitivity of holding cost has not yet been exhaustively covered within the available literature. This research will be motivated by the need to fill this gap in the integration by creating a hybrid inventory optimization model using AI and Fuzzy. The suggested model is smart enough to combine AI-powered demand prediction with the use of LSTM networks and fuzzy defuzzification with cost modelling based on deterioration. The integration improves the accuracy of the decisions, minimizes the uncertainty of the computations and offers a strong structure of the minimization of the overall inventory cost in the difficult and real-world systems.

3. Methodology

The proposed study follows a structured methodology that integrates AI-based forecasting, fuzzy logic, and inventory cost optimization within a unified framework.

3.1 Problem Conceptualization and Assumptions

The study deals with an inventory management issue that is of real-world situation where the demand of the products is not known and the value of the products reduces as they get older through depreciation. Standard deterministic models have a tendency to neglect two important issues:

1.        The uncertainty in demand and inaccurate determination of the order quantities, which results in inaccurate order quantity decisions.

2.        Dynamic holding costs which rise when deterioration is large.

To address these shortcomings, the model conceptualizes an inventory system where the demand is initially forecasted with the help of AI-based forecasting and then subjected to a fuzzy variable to indicate uncertainty. The holding cost is supposed to be dependent on the degree of deterioration which forms a realistic relationship between the perishability of the product and the storage costs. The assumptions used in the formulation of the model are realistic and practical in nature and are in line with the characteristics of deteriorating items and actual market environments.

3.2 AI-Based Forecasting Using LSTM

A Long Short-Term Memory (LSTM) neural network is used to forecast the demand. The LSTM model is also an application of the recurrent neural network, but it has been found to be useful in sequential time-series data due to its ability to store long-term dependencies. The model is trained using the historical demand values after which it is applied to make predictions of the future demand of the next time period. It becomes familiar with complicated, nonlinear demand trends that cannot be represented by standard forecasting schemes. The AI-predicted demand value is the output of this stage and the fuzzy modelling has this value as the base input.

3.3 Fuzzy Representation of the Predicted Demand

Given that the AI predictions are prone to uncertainty, the predicted demand is fuzzily expressed to consider vagueness in actual market conditions like market changes, consumer behaviour, and environmental conditions. The lower, middle and upper ranges of demand estimations are defined by a triangular fuzzy representation. This will enable the model to be more realistic in expressing the inherent uncertainty.

3.4 Defuzzification

The fuzzy demand is converted into a single crisp value using the centroid method of defuzzification. This defuzzified demand serves as an effective and representative value of demand for the cost optimization stage, balancing both optimistic and pessimistic scenarios.

3.5 Model Formulation and Optimization

The next step will be to develop the model of inventory costs based on deterioration, holding cost, and ordering cost. The holding cost is considered a variable which is also dependent on the rate of deterioration, which makes the model more realistic of the real storage conditions. The model uses optimization methods to get the best inventory cycle time and order quantity that gives the least total cost. This is done numerically as the optimization

process is conducted using computational tools. The implementation of the methodology is through Python programming language. Forecasting, computation, and visualization are done with the help of important packages like TensorFlow, NumPy, Pandas, SciPy and Matplotlib.

3.6 Sensitivity Analysis

Sensitivity analysis is performed to assess how variations in key parameters affect the total cost. Parameters such as deterioration rate, holding cost sensitivity, fuzzy spread, and AI-predicted demand are altered by ±10% and ±20% to study the impact on overall performance. This ensures that the model is robust, stable, and reliable under different operating conditions.

4. Assumptions and Notations

4.1 Assumptions:

To construct the inventory model under a complex environment, the following realistic assumptions are made:

1.     The system deals with a single type of deteriorating product with no replenishment shortages.

2.    Demand rate  is not constant but fuzzy in nature; it is represented by a triangular fuzzy number , where denotes the most likely demand and represent optimistic and pessimistic estimates, respectively.

3.     The holding cost (h) is deterioration-dependent, increasing proportionally with deterioration rate :                   

where is the base holding cost, and is the sensitivity coefficient.

4.      Deterioration follows an exponential pattern, with constant rate .

5.        Lead time is zero, and replenishment is instantaneous.

6.        No shortages are allowed, and inventory level becomes zero at the end of each cycle.

7.        The planning horizon is infinite, and costs are measured per unit time.

4.2 Notation Table

5.  Mathematical Formulation of the model

5.1 Inventory Level

Let denote the inventory level at time during the replenishment cycle .
The differential equation representing inventory depletion due to demand and deterioration is:

Integrating this equation over the interval :

and the total inventory quantity at the beginning of the cycle is obtained as:

5.2 Ordering Cost

5.3 Holding Cost

Where: 

Substituting ,

5.4 Deterioration Cost

Deteriorated items per cycle are given by:      

thus,

5.5 Total Cost per Cycle

Substituting all three:

5.6 Average Total Cost (ATC) Function

=

The average total cost function ATC(T) explicitly depends on all system parameters including ordering cost , base holding cost , deterioration rate , deterioration cost , sensitivity parameter , and effective defuzzified demand . Therefore, the optimization problem can be mathematically formulated as:

Minimize

subject to .

The objective is to determine such that:

Since a closed-form solution is complex, the optimal value of is obtained numerically using nonlinear optimization (e.g., scipy.optimize.minimize_scalar () in Python).

5.7 AI–Fuzzy Demand Representation

The uncertain demand is first predicted through an AI based forecasting system, such as an LSTM neural network, which yields a crisp forecast . To incorporate uncertainty, a triangular fuzzy number is assigned, where and represent lower and upper tolerance limits. In the proposed model, demand uncertainty is represented using an asymmetric triangular fuzzy number defined as . Here, 0.85 and 1.20 are used as the left and right spread factors, respectively. The value 0.85  corresponds to the pessimistic (lower support) limit, indicating that the actual demand may fall up to 15% below the AI forecast. Conversely, 1.20  denotes the optimistic (upper support) limit, allowing the demand to rise up to 20% above the forecast. These spread factors, also known as uncertainty spread parameters, quantify the range of possible deviations around the LSTM predicted demand and help capture real world asymmetric uncertainty. The choice of these values aligns with recent fuzzy inventory literature, where 10–25% variability around the expected demand is commonly adopted to represent uncertainty realistically.

The defuzzified AI based effective demand is obtained using the centroid method:

Hence, the AI Fuzzy integration produces a smooth and robust estimate of demand used for further optimization.

6. Optimization Approach

The optimization approach establishes the procedure used to determine the most cost-efficient replenishment policy for deteriorating items under fuzzy and AI-based demand uncertainty. After formulating the total cost structure, the objective is to identify the optimal replenishment cycle time that minimizes the average total cost. The following subsections describe the optimization logic, mathematical behaviour of the cost function, and the computational method used to obtain the optimal solution.

6.1 Objective of Optimization

The objective of the optimization process is to determine the replenishment cycle time that minimizes the average cost of operating the inventory system. This optimal value ensures:

·                  minimum ordering frequency,

·                  balanced holding duration,

·                  reduced deterioration impact, and

·                  lowest combined inventory expenditure.

By using AI-generated demand forecasts, the model aligns real-time data with cost-effective inventory policies.

6.2 Nature of the Cost Function

The average total cost function formulated in this study is nonlinear due to the presence of exponential deterioration and time dependent inventory behaviour. Nonlinearity makes analytical minimization inconvenient, but the function exhibits convexity, which is an essential property for optimization. Convexity ensures:

·       there is a unique global minimum,

·       the optimal cycle time is stable,

·       numerical optimization techniques work reliably.

This guarantees that any decline in cycle time beyond the optimal value increases cost, and any increase in cycle time beyond that point also increases cost.

6.3 Decision Variable and Output Parameters

The sole decision variable in the optimization process is:

·       Cycle time (T) - duration between two consecutive replenishment orders.

Once the optimum value of T is identified, the model subsequently computes:

·       Optimal order quantity,

·       Total cost per replenishment cycle, and

·       Minimal average total cost.

These outputs form the backbone of inventory policy decisions.

6.4 Optimization Technique Used

Since the average total cost function cannot be minimized using simple calculus due to its nonlinear exponential form, the study applies numerical optimization techniques. method used is: Bounded Scalar Minimization using minimize_scalar ().

·       The cycle time is searched within a realistic interval.

·       The algorithm evaluates the cost function iteratively within this range.

·       The searching procedure concludes at the value where the cost becomes minimum.

This method is well-suited for convex cost structures and gives fast, accurate, and stable results.

6.5 Step by Step Optimization Procedure

The optimization process follows these clear steps:

1.        Input all inventory parameters (ordering cost, holding cost, deterioration rate, fuzzy demand, etc.).

2.        Construct the total cost function using the cost components derived earlier.

3.        Define a feasible search interval for the cycle time.

4.        Run the numerical optimization routine to explore where the minimum cost occurs.

5.        Identify the optimal cycle time that minimizes the cost.

6.        Calculate optimal order quantity using the obtained cycle time.

7.        Compute the minimum average total cost for that optimal cycle.

7. Numerical Illustration

Step 1: LSTM Forecast

The historical demand data for the last ten periods was considered as:
.

The historical demand data used in this study represents realistic demand patterns observed in perishable food retail supply chains. Such nonlinear fluctuations commonly arise in supermarket inventory systems due to seasonal variation, short-term demand shocks, and market uncertainty. This time-series was used as input to the LSTM neural network, which captures nonlinear and short-term variations more effectively than traditional statistical models. An LSTM model was trained to capture nonlinear temporal patterns. The trained network predicted the next-period demand as = 110.68 units, which was subsequently used for fuzzy defuzzification and cost optimization.

Step 2: Fuzzy Representation

Step 3: Defuzzification

Defuzzified Demand  = 111.60

Step 4: Optimization

Let   , ,  , ,  

Using these data through Python software, optimal values are calculated as:

Optimal Cycle Time  = 0.65

Optimal Order Quantity  = 72.24

Minimum ATC =  = 152.80

8. Sensitivity Analysis

Sensitivity analysis was performed to evaluate the robustness and stability of the proposed fuzzy–AI inventory model with respect to its key parameters. Four major parameters were varied:
(1) defuzzified demand

(2) holding cost

(3) deterioration cost  

(4) deterioration rate .

Each parameter was perturbed by –20%, –10%, 0%, +10%, and +20%, and for every perturbed value, the optimal cycle time and the corresponding minimum total cost  were recalculated. This approach ensures economically meaningful sensitivity results because the model optimizes the replenishment strategy for every parameter change.

Following observations are derived: -

1. Sensitivity with Respect to Fuzzy Demand

The results show that total cost increases as demand increases. A positive change of +10% and +20% in significantly raises the optimal total cost due to the higher level of required inventory and greater exposure to deterioration. Conversely, a reduction in demand (–10%, –20%) yields lower optimal cost because less inventory is held and the model operates with shorter effective cycles. Thus, the model shows direct proportionality between demand and optimal cost.

2. Sensitivity with Respect to Holding Cost

As expected, an increase in the holding cost component results in a steady rise in the total cost. When increases, the inventory becomes more expensive to store, and even though the optimal cycle time slightly adjusts, the holding cost dominates, causing overall cost escalation. When decreases (–10%, –20%), the total cost declines accordingly. This indicates that holding cost is a strongly influential parameter in cost behaviour

3. Sensitivity with Respect to Deterioration Cost

Variations in the deterioration cost show a moderate impact on total cost. An increase in leads to higher inventory costs since each deteriorated unit contributes a larger penalty. Conversely, lower values reduce the total cost but not as drastically as demand or holding cost changes.
The model exhibits a balanced and predictable response to deterioration cost variations.

4. Sensitivity with Respect to Deterioration Rate

An interesting and economically meaningful trend is observed when the deterioration rate is varied. Unlike most classical EOQ-type models, an increase in the deterioration rate results in a decrease in the minimum total cost in this study.

This behaviour occurs because the sensitivity analysis recalculates a new optimal cycle time for every value of . When the deterioration rate increases, the system compensates by choosing shorter replenishment cycles, which:

·       reduce the average inventory level,

·       significantly lower holding costs, and

·       limit total deterioration over time.

The reduction in holding cost outweighs the increase in deterioration rate, resulting in net cost reduction.

In all cases, the model remains stable, and total cost changes smoothly without erratic fluctuations. The results validate that the proposed inventory framework is robust, well-behaved, and managerially reliable.

9. Results and Discussion

The LSTM-based forecasting model, trained on the recent demand history, predicted a next-period demand of 110.68 units, which was converted into an effective defuzzified demand of 111.60 units using an asymmetric fuzzy representation. Using this demand, the proposed deterioration-dependent inventory model produced an optimal cycle time of 0.65, an optimal order quantity of 72.24 units, and a minimum average total cost of 152.80 per cycle time. The shape of the ATC curve was smooth and its global minimum, which proves the stability and model feasibility. Sensitivity analysis revealed that the demand and holding cost have the most impact on the total cost, and deterioration cost has a moderate impact. Interestingly, an increase in the deterioration rates decreased total cost by decreasing the optimal cycles and decreasing the average inventory as a demonstration of the adaptive behavior of the proposed model to uncertainty and perishability. All in all, the findings confirm that the model is stable, responsive, and is appropriate to be used in practice to make inventory decisions.

10. Application

The suggested AI-fuzzy inventory model can be widely applied in practice to various industrial industries that have to face unpredictable demand and deterioration-sensitive products. Perishable food supply chains such as fruits, vegetables, dairy, bakery products and frozen foods demand is not stable but changes regularly and products quality decays with time. The combination of LSTM forecasting with the fuzzy uncertainty modelling allows the managers to predict the demand of near future with greater precision, whereas the deterioration-dependent holding cost formulation allows reducing wastage and refrigeration cost by dynamically adjusted replenishment cycles. Likewise, in the pharmaceutical and healthcare industry, where medical drugs, vaccines, diagnostic chemicals, and blood bags have very limited shelf-life, the model helps in efficient inventory management by minimizing expiry wastage and guaranteeing a good supply of important products at the right time. The agricultural/cold-chain logistics industry will also be able to gain a lot, since such commodities as grains, pulses, seeds, flowers, and export perishables need to be kept in controlled storage and replenished on a regular basis to preserve its quality; the model offers the best ordering intervals that will help save the costs of energy and degradation losses. The model can be used in retail and supermarket operations by having more precise replenishment scheduling and inventory rotation to minimize the cases of overstocking and stock-out, as these businesses experience daily fluctuations in consumer demand of packaged food, beverages, and other essential products. Moreover, the framework is very much applicable to the e-commerce and quick-commerce models that are based on short-term and unstable demand trends; the AI-based forecasting element enhances promptness, whereas the optimization unit modulates the quantities of orders to minimize returns, spoilage, and storage expenses. The model can assist industrial sectors that work with degraded raw materials, such as chemicals, paints, adhesives, solvents, and industrial oils, to decide on the most economical procurement quantities, and prevent losses caused by evaporation, oxidation, or chemical decomposition. In addition, hospitals, diagnostic centres and blood banks that hold high value, perishable goods of reagents, sterilization and blood components can use the model as a dependable means to maintain good stock without incurring too much holding cost. On the whole, the model can be applied in any setting where the products get depreciated with time, demand can be unpredictable, and cost-effectiveness based on the smart AI-enhanced decision-making is essential.

11.Conclusion
This study presents a comprehensive and robust inventory framework that integrates AI-based demand forecasting, asymmetric fuzzy modelling, and deterioration-dependent cost optimization for complex systems. The LSTM-based forecasting module successfully captures nonlinear patterns in historical demand and provides a reliable forecast, which is further refined through fuzzy logic to account for uncertainty and real-world variability. The resulting effective demand improves the accuracy and realism of the inventory decisions. The mathematical formulation incorporates deterioration-dependent holding costs, reflecting the practical situation where higher deterioration makes inventory more expensive to handle and store. Optimization results confirm the existence of a unique optimal cycle time that minimizes the overall cost. The model produces feasible and economically interpretable values for the optimal order quantity, optimal cycle time, and minimum total cost. Sensitivity analysis reveals that the proposed model is highly stable and reacts in an economically meaningful way to parameter fluctuations. Demand and holding cost exert the strongest influence on total cost, whereas deterioration cost contributes moderately. Interestingly, an increase in the deterioration rate results in a reduction in the optimal total cost, as the model adapts by reducing the cycle time and thereby lowering average inventory levels. This behaviour highlights the adaptive strength of the model and its ability to maintain cost efficiency even under adverse deterioration conditions. Overall, the integration of AI forecasting, fuzzy uncertainty modelling, and deterioration-based optimization results in a highly effective inventory system suitable for modern supply chains dealing with perishability and uncertainty. The findings demonstrate that the proposed model is practical, mathematically sound, and managerially beneficial, offering improved decision-making capabilities for organizations operating in dynamic and uncertain environments.

Declarations

Funding: This research received no external funding.

Conflict of Interest: The Authors declare that there is no conflict of interest.

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