Annals of the Academy of Romanian Scientists

Series on Engineering Sciences

ISSN 2066-6950

Volume 18, Number 1/2026

36

BAYESIAN INVENTORY CONTROL UNDER UNCERTAIN

DEMAND: A DATA-DRIVEN LEARNING APPROACH

Marcel ILIE¹and Augustin SEMENESCU²

Rezumat. Gestionarea eficientă a stocurilor în condiții de incertitudine rămâne o

provocare centrală în sistemele lanțului de aprovizionare, în special atunci când cererea

este stocastică, nestaționară sau parțial observată. Metodele clasice de control al

stocurilor se bazează pe ipoteze probabilistice fixe care adesea nu reușesc să surprindă

dinamica cererii în evoluție. Această lucrare dezvoltă un cadru Bayesian de Control al

Stocurilor care integrează învățarea probabilistică a cererii cu luarea deciziilor

secvențiale. Abordarea propusă actualizează distribuțiile cererii în timp real folosind

inferența Bayesiană, permițând politici de comandă adaptive care răspund la informații

noi. Pentru a evalua performanța, modelul Bayesian este comparat cu politicile

tradiționale de inventar bazate pe prognoză, inclusiv prognoza statistică bazată pe ARIMA

și predicția cererii prin învățare profundă bazată pe LSTM. Rezultatele simulării

demonstrează că abordarea Bayesiană reduce constant costul total al stocurilor,

îmbunătățește nivelurile de servicii și prezintă o robustețe superioară în scenarii de

incertitudine ridicată a cererii și disponibilitate scăzută a datelor. Constatările evidențiază

avantajele combinării inferenței probabilistice cu optimizarea stocurilor în sistemele

moderne ale lanțului de aprovizionare.

Abstract. Effective inventory management under uncertainty remains a central challenge

in supply chain systems, particularly when demand is stochastic, non-stationary, or

partially observed. Classical inventory control methods rely on fixed probabilistic

assumptions that often fail to capture evolving demand dynamics. This paper develops a

Bayesian Inventory Control framework that integrates probabilistic demand learning with

sequential decision-making. The proposed approach updates demand distributions in real

time using Bayesian inference, enabling adaptive ordering policies that respond to new

information. To evaluate performance, the Bayesian model is compared with traditional

forecasting-driven inventory policies, including ARIMA-based statistical forecasting and

LSTM-based deep learning demand prediction. Simulation results demonstrate that the

Bayesian approach consistently reduces total inventory cost, improves service levels, and

shows superior robustness under high demand uncertainty and low data availability

scenarios. The findings highlight the advantages of combining probabilistic inference with

inventory optimization in modern supply chain systems.

Keywords: Bayesian inventory control; stochastic inventory management; (s,S) policy;

demand forecasting; uncertainty quantification; machine learning forecasting models

DOI 10.56082/annalsarscieng.2026.1.36

¹Associate. Prof. Ph.D. Georgia Southern University, 1332 Southern Dr. Statesboro GA 30458,

USA, *Corresponding author: milie@georgiasouthern.edu

²Prof. National Science and Technology University Politehnica Bucharest, Spl. Independentei

313, Bucharest, Romania, augustin.semenescu@upb.ro

36

Real-time extraction of neuromorphic features for nuclear and industrial security

37

1. Introduction

Inventory control is a foundational problem in operations research and

supply chain management, where firms must carefully balance ordering costs,

holding costs, and the risks associated with stockouts. Effective inventory policies

are essential for maintaining service levels while minimizing total operational cost,

particularly in competitive and uncertain market environments. Classical inventory

models, such as the Economic Order Quantity (EOQ) framework and (s, S) policies,

have long served as standard tools for decision-making under uncertainty [1–4].

These models, however, typically rely on strong assumptions, including known and

stationary demand distributions, constant lead times, and stable system dynamics

[1–3]. While analytically tractable, such assumptions are often violated in real-

world supply chains [7].

In practice, demand processes are rarely stationary and are frequently

influenced by structural breaks, seasonality, external shocks, and evolving

consumer behavior [12]. Furthermore, decision-makers often operate under

conditions of incomplete information, especially in the context of new product

introductions or rapidly changing markets [9]. These challenges limit the

applicability of classical inventory models and motivate the development of

adaptive and data-driven approaches that can learn from observed demand over

time [10].

Bayesian methods provide a principled and coherent framework for

addressing these limitations [16–17]. By treating unknown demand parameters as

random variables rather than fixed but unknown constants, Bayesian inventory

control enables the explicit representation of uncertainty in model parameters [8].

As new demand observations become available, prior beliefs are updated through

Bayes’ theorem, resulting in posterior distributions that reflect improved

knowledge of the underlying demand process [16]. This iterative learning

mechanism allows decision-makers to continuously refine their understanding of

demand while simultaneously optimizing inventory decisions [17].

A key advantage of Bayesian inventory control lies in its integration of

learning and decision-making within a unified framework [8]. Unlike traditional

forecasting pipelines, which typically separate demand prediction from operational

optimization, Bayesian approaches directly incorporate uncertainty into the

decision process [15]. As a result, ordering policies are dynamically adjusted based

on posterior distributions rather than point estimates, enabling more robust

performance under uncertainty [10]. This is particularly valuable in environments

with limited historical data, high volatility, or rapidly evolving demand structures

[9].

Recent advances in machine learning have further enhanced the ability to

model complex demand patterns. Techniques such as ARIMA-based time series

models and deep learning architectures like Long Short-Term Memory (LSTM)

38

Marcel Ilie and Augustin Semenescu

networks have demonstrated strong performance in demand forecasting tasks [11–

13]. However, these approaches are often primarily predictive in nature and remain

decoupled from inventory decision-making frameworks [14-20]. In many cases,

forecasts generated by such models are used as inputs to separate optimization

routines, which may not fully capture forecast uncertainty or its impact on

operational decisions [12].

This paper aims to bridge this gap by comparing Bayesian decision-making

approaches with both classical statistical forecasting and modern deep learning

methods within a unified inventory control setting. In particular, we evaluate how

different modeling paradigms—Bayesian updating, ARIMA-based forecasting, and

LSTM-based learning—affect inventory performance metrics such as total cost,

service level, and responsiveness to demand changes. By integrating forecasting

and optimization perspectives, this study highlights the advantages and limitations

of each approach in dynamic and uncertain environments and demonstrates the

practical value of Bayesian learning for adaptive inventory control.

2. Mathematical modeling and algorithms

2.1 Demand Model

Let 퐷_푡denote the demand in period ꢀ = 1,2, … , 푇. We assume demand

follows a stochastic process parameterized by an unknown vector 휃. A common

and tractable assumption is that demand is conditionally independent given 휃:

퐷_푡∣ 휃 ∼ 풟(휃)

(1)

where 풟(휃)may represent a Poisson, Normal, or other suitable demand distribution

depending on the application context. For instance, in the case of Gaussian demand:

퐷_푡∣ 휇, 휎²∼ 풩(휇, 휎²)

(2)

The unknown parameters 휃 = (휇, 휎²) are treated as random variables in the

Bayesian framework.

2.2 Prior Distribution

Before observing any demand data, the decision-maker specifies prior distribution

over the unknown demand parameters:

푝(휃)

(3)

For conjugate analysis (Gaussian demand with unknown mean and known

variance), a typical prior is:

휇 ∼ 풩(휇₀, 휏²)

(4)

0

Real-time extraction of neuromorphic features for nuclear and industrial security

39

This prior encodes initial beliefs based on historical data, expert judgment, or

market information.

2.3 Bayesian Updating

As demand observations become available, beliefs are updated using Bayes’

theorem. After observing demand history 퐷_1:푡= {퐷₁, … , 퐷_푡}, the posterior

distribution is:

ꢁ(ꢂ ∣ꢄ)ꢁ(ꢄ)

1:ꢃ

푝(휃 ∣ 퐷_1:푡) =

(5)

ꢁ(ꢂ

1:ꢃ

)

For Gaussian demand with known variance, the posterior mean updates as:

−2

0

−2

ˉ

ꢂ

ꢃ

ꢅ

ꢆ +푡ꢇ

−2

+푡ꢇ

0

휇_푡=

(6)

(7)

−2

ꢅ

where:

1

푡

ˉ

∑

퐷_푡=

_푖=1퐷_푖

푡

This recursive structure allows continuous learning as new data arrives.

,

(

)

2.4. Inventory System and 풔 푺 Policy

We consider a periodic-review inventory system with instantaneous

replenishment. Let 퐼_푡denote the inventory level at the beginning of period ꢀ. The

system evolves as:

퐼_푡+1= 퐼_푡+ 푄_푡− 퐷_푡

(8)

where 푄_푡is the order quantity.

,

(

)

Under an 푠 푆 policy:

• If 퐼_푡< 푠, order up to level 푆

• Otherwise, no order is placed

•

푆 − 퐼_푡,

퐼_푡< 푠

퐼_푡≥ 푠

푄_푡= {

(9)

0,

2.5 Bayesian Optimal Decision Rule

In the Bayesian setting, optimal order decisions are based on the posterior

predictive distribution of demand:

푝(퐷_푡+1∣ 퐷_1:푡) = ∫ 푝(퐷_푡+1∣ 휃)푝(휃 ∣ 퐷_1:푡)푑휃

(10)

40

The expected demand under uncertainty is:

피[퐷_푡+1∣ 퐷_1:푡] = ∫ 퐷_푡+1ꢀ푝(퐷_푡+1∣ 퐷_1:푡)ꢀ푑퐷_푡+1

The 푠 푆 parameters are dynamically updated by minimizing expected total cost:

Marcel Ilie and Augustin Semenescu

(11)

,

(

)

(푠^∗, 푆^∗) = arg min _ꢈ,ꢉꢁ피[ℎ(퐼_푡)⁺+ 푏(퐼_푡)⁻+ 퐾ퟏ_{ꢊ >0}∣ 퐷_1:푡

(12)

ꢃ

where:

•

ℎ: holding cost

푏: shortage (backorder) cost

퐾: fixed ordering cost

+

−

(

) ( )

푥) = 푚푎푥(푥, 0 , 푥) = 푚푎푥(−푥, 0

2.6 Cost Function

The total expected cost per period is defined as:

퐶_푡= ℎmax (퐼_푡, 0) + 푏max (−퐼_푡, 0) + 퐾ퟏ_{ꢊ >0}

(13)

(14)

ꢃ

The objective is to minimize long-run average cost:

ꢋ

1

min ꢁlim _ꢋ→∞

∑

_푡=1피[ 퐶_푡]

ꢋ

2.7 Integration with Learning

Unlike classical inventory systems where parameters are fixed, the

Bayesian framework continuously adapts:

1. Observe demand 퐷_푡

2. Update posterior 푝(휃 ∣ 퐷_1:푡)

3. Recompute predictive demand distribution

,

(

)

4. Update 푠 푆 policy

5. Execute ordering decision 푄_푡

This closed-loop structure enables real-time adaptation to non-stationary demand

environments.

3. Results and Discussion

3.1 Cumulative Inventory Cost Comparison

The cumulative cost trajectories for the Bayesian, ARIMA-based, and

LSTM-based inventory policies are illustrated in Figure 1. Across the entire

simulation horizon, the Bayesian inventory control strategy consistently achieves

the lowest cumulative cost. This improvement is not merely marginal but becomes

Real-time extraction of neuromorphic features for nuclear and industrial security

41

increasingly pronounced as time progresses, indicating a compounding effect of

adaptive learning.

Figure 1. Cumulative Inventory Cost Comparison

A key observation is that the performance gap between Bayesian and the

benchmark models widens over time. Early in the simulation, all three approaches

exhibit relatively similar cost behavior, as limited historical data constrains the

effectiveness of learning-based adjustments. However, as additional demand

observations become available, the Bayesian model progressively refines its

posterior distribution of demand, leading to increasingly accurate order decisions.

In contrast, the ARIMA-based approach relies on rolling statistical estimates that

assume local stationarity, which limits its ability to adapt to structural shifts or high-

variance demand realizations. Similarly, the LSTM-based model, while capable of

capturing nonlinear temporal patterns, is sensitive to noise and requires large

datasets to stabilize its predictions. This results in persistent over- or under-

ordering, which accumulates into higher total cost. Figure 1 therefore highlights a

fundamental advantage of Bayesian inventory control: its ability to reduce decision

42

Marcel Ilie and Augustin Semenescu

error through continuous probabilistic updating, rather than relying on fixed or

purely predictive mappings.

3.2 Service Level Performance Over Time

Service level performance, defined as the proportion of demand satisfied

without stockouts, is presented in Figure 2. The Bayesian policy demonstrates both

higher average service levels and significantly lower variance compared to the

benchmark approaches.

Figure 2. Service Level Over Time

The stability of the Bayesian service level curve is particularly important

from an operational perspective. In supply chain systems, variability in service

levels is often as critical as the average performance, since fluctuations directly

translate into customer dissatisfaction and supply uncertainty. The Bayesian model

maintains a consistently high service level by dynamically adjusting order

quantities in response to updated posterior demand distributions. By contrast, the

ARIMA-based policy performs adequately under stable demand conditions but

shows degradation when demand variability increases. This is primarily due to its

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43

reliance on fixed-window statistical structure, which reacts slowly to abrupt

changes in demand behavior. The LSTM-based approach exhibits even higher

variability in service levels, reflecting its sensitivity to short-term fluctuations and

potential overfitting to recent demand patterns. Overall, Figure 2 demonstrates that

Bayesian inventory control not only improves average service performance but also

significantly enhances operational robustness under stochastic demand conditions.

3.3 Bayesian Demand Learning (Posterior Update Visualization)

Figure 3 illustrates the evolution of the Bayesian demand estimate over time.

The estimated mean demand converges progressively toward the true demand level

as additional observations are incorporated into the posterior distribution.

Figure 3. Bayesian Demand Mean Update Over Time

In the early stages of the simulation, the model exhibits relatively high

uncertainty, reflecting limited data availability and stronger influence from the prior

distribution. As time progresses, the posterior distribution becomes increasingly

44

Marcel Ilie and Augustin Semenescu

dominated by observed demand data, resulting in a steady reduction in estimation

error. This learning behavior is a key distinguishing feature of Bayesian inventory

control. Unlike ARIMA or LSTM models, which produce point forecasts, the

Bayesian framework explicitly maintains and updates a full probability distribution

over demand uncertainty. This allows the model not only to estimate expected

demand but also to quantify uncertainty, which is directly incorporated into

ordering decisions. The convergence pattern observed in Figure 3 confirms that

Bayesian updating provides a theoretically consistent mechanism for reducing

uncertainty over time, which directly translates into improved inventory decisions.

3.4 Forecast Accuracy Comparison

Figure 4 presents the evolution of absolute forecast errors for the Bayesian,

ARIMA-based, and LSTM-based demand estimation models.

Figure 4. Forecast Accuracy Comparison

The results show a clear separation in predictive performance across the

three approaches. The Bayesian estimator demonstrates a steadily decreasing error

trajectory over time. This behavior reflects the continuous refinement of the

posterior distribution as additional demand observations are incorporated. Early-

stage errors are relatively higher due to limited information; however, the model

Real-time extraction of neuromorphic features for nuclear and industrial security

45

rapidly converges toward the true demand mean, resulting in stable and low

prediction error in later periods. In contrast, the ARIMA-based approach exhibits

moderate but persistent error levels. While it performs reasonably well under

locally stationary demand, it struggles to adapt to stochastic fluctuations and

structural variability. The LSTM-based model shows the highest variability in

forecast error, particularly in early and mid-stage periods. This instability is

attributed to its sensitivity to short-term noise and its dependence on sufficient

training data to stabilize learning. Overall, Figure 4 confirms that Bayesian learning

provides superior long-term forecast accuracy and more stable convergence

behavior compared to both statistical and deep learning benchmarks.

3.5 Inventory Level Dynamics (Figure 5)

Figure 5 illustrates the inventory level trajectories under the three control

policies. The dynamics of inventory positions provide important insight into system

stability beyond cost and forecast accuracy metrics.

Figure 5. Inventory Level Dynamics

The Bayesian inventory policy produces the smoothest and most stable

inventory trajectory. Adjustments in inventory levels occur gradually as the

46

Marcel Ilie and Augustin Semenescu

posterior demand estimate evolves, preventing excessive oscillations in ordering

decisions. This indicates a more balanced response to demand uncertainty. The

ARIMA-based policy exhibits moderate oscillatory behavior, characterized by

delayed adjustments to demand changes. This lag leads to periodic overstocking

and understocking cycles. The LSTM-based policy shows the highest variability in

inventory levels, reflecting sensitivity to short-term prediction errors and frequent

corrective adjustments. These results demonstrate that Bayesian inventory control

not only improves cost efficiency but also enhances system stability by reducing

volatility in inventory decisions.

3.6 Stockout Frequency Analysis

Figure 6 compares the total number of stockout events across the three

methods. Stockouts represent critical service failures in supply chain systems and

directly impact customer satisfaction.

Figure 6. Stockout Frequency Analysis

The Bayesian approach results in the lowest number of stockout

occurrences. This is due to its ability to incorporate predictive uncertainty into

ordering decisions, allowing it to maintain more robust safety stock levels

Real-time extraction of neuromorphic features for nuclear and industrial security

47

dynamically. The ARIMA-based method exhibits a moderate stockout frequency,

reflecting its limited adaptability to sudden demand variations. The LSTM-based

method experiences the highest number of stockouts, particularly in volatile

demand periods, where forecasting errors propagate directly into inventory

shortages. These findings highlight that Bayesian inventory control provides a more

reliable service-level guarantee under uncertainty compared to purely predictive

approaches.

3.7 Cost Variability and Risk Stability

Figure 7 presents the standard deviation of periodic costs for each inventory

policy, serving as a measure of financial risk and operational stability.

Figure 7. Cost Variability

The Bayesian approach exhibits the lowest cost variability among all

methods. This indicates not only lower average cost but also more predictable

financial performance over time. Reduced variance is particularly important in

supply chain planning, where cost stability is often as critical as cost minimization.

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Marcel Ilie and Augustin Semenescu

The ARIMA-based policy shows moderate variability, reflecting its dependence on

rolling-window estimates that react slowly to demand shifts. The LSTM-based

model exhibits the highest cost variance, driven by fluctuating prediction errors and

over-adjustment behavior. These results demonstrate that Bayesian inventory

control significantly improves risk stability in addition to cost efficiency.

3.8 Bayesian Uncertainty Reduction

Figure 8 illustrates the evolution of posterior variance over time, capturing

the learning dynamics of the Bayesian model.

Figure 8. Bayesian Uncertainty Reduction

The results show a clear and monotonic decrease in uncertainty as more

demand observations become available. Initially, the posterior variance is relatively

high due to limited data and stronger reliance on prior assumptions. However, as

the system accumulates information, the posterior distribution becomes

increasingly concentrated around the true demand value. This reduction in

uncertainty is a key mechanism underlying the improved performance observed in

Real-time extraction of neuromorphic features for nuclear and industrial security

49

previous figures. By explicitly quantifying and reducing uncertainty over time, the

Bayesian framework enables progressively more accurate and risk-aware inventory

decisions. Unlike ARIMA and LSTM models, which provide point estimates

without explicit uncertainty representation, the Bayesian approach directly

integrates uncertainty into the decision-making process.

4 Discussion

The comparative analysis across all experiments consistently demonstrates

the superiority of Bayesian inventory control over ARIMA- and LSTM-based

benchmark approaches. Three central insights emerge from the results regarding

cost efficiency, operational stability, and uncertainty handling.

4.1 Learning-driven improvement in cost efficiency

Bayesian inventory control continuously updates the demand distribution

through probabilistic learning, enabling progressive refinement of ordering

decisions. This adaptive mechanism systematically reduces both overstocking and

understocking events. As a result, it achieves lower cumulative inventory costs

compared to ARIMA and LSTM-based methods, which rely on either static

assumptions or decoupled forecasting pipelines. Notably, the cost advantage of the

Bayesian approach increases over time, highlighting its long-term efficiency in

dynamic environments.

4.2 Operational stability and service level consistency

The Bayesian framework provides significantly more stable performance in

terms of service levels and inventory trajectories. Unlike benchmark models, which

exhibit greater variability due to forecasting errors and sensitivity to demand noise,

the Bayesian policy maintains consistent service levels even under high demand

volatility. This stability is critical in practical supply chain settings, where reliability

often carries equal importance to cost minimization.

4.3 Explicit uncertainty modeling and risk-aware decisions

A key advantage of the Bayesian approach lies in its explicit representation

of uncertainty. Instead of relying on single-point forecasts, it integrates the full

posterior distribution of demand into the decision-making process. This leads to

more informed and risk-aware ordering policies, particularly in data-scarce or

highly stochastic environments. By directly incorporating uncertainty into

optimization, the Bayesian framework reduces decision risk and improves

robustness.

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Marcel Ilie and Augustin Semenescu

4.4 Overall interpretation

Collectively, the results demonstrate that integrating probabilistic learning

directly into inventory control yields a fundamental improvement over traditional

forecast-then-optimize frameworks. While ARIMA and LSTM models may

improve predictive accuracy, they remain structurally separated from the decision

process and do not explicitly account for uncertainty in optimization. In contrast,

Bayesian inventory control unifies forecasting and decision-making within a single

coherent probabilistic framework. This integration leads to improved cost

efficiency, enhanced service level stability, and greater robustness under demand

uncertainty. Across all experimental figures, a consistent pattern is observed:

Bayesian inventory control outperforms ARIMA- and LSTM-based approaches in

accuracy, stability, cost performance, and risk management. These findings

strongly support the adoption of Bayesian methods in modern inventory systems,

particularly in environments characterized by volatile demand and limited historical

data.

5. Conclusions

This study developed and evaluated a Bayesian Inventory Control

framework for managing stochastic demand under uncertainty. Unlike classical

inventory policies that rely on fixed demand distributions or separate forecasting

and optimization stages, the proposed approach integrates demand learning and

decision-making within a unified probabilistic framework. By continuously

updating the posterior distribution of demand as new observations become

available, the model enables adaptive ordering decisions that improve over time.

The numerical results demonstrate that the Bayesian approach consistently

outperforms benchmark methods, including ARIMA-based forecasting and LSTM-

based predictive models. In terms of total inventory cost, the Bayesian policy

achieves the lowest cumulative cost across all simulated scenarios, with

performance advantages becoming more pronounced over longer time horizons.

This improvement is primarily driven by the model’s ability to progressively

reduce demand uncertainty and avoid systematic ordering errors.

In addition to cost efficiency, the Bayesian framework also delivers superior

service level performance. The results show higher and more stable service levels

compared to the benchmark methods, particularly under high demand variability.

This stability is a critical operational advantage, as it reflects more reliable

fulfillment performance and reduced risk of stockouts in uncertain environments.

A key finding of this study is the importance of explicit uncertainty

modeling. While ARIMA and LSTM approaches can provide accurate point

forecasts under certain conditions, they do not directly incorporate predictive

uncertainty into the decision process. In contrast, Bayesian inventory control

explicitly represents uncertainty through posterior distributions, allowing for more

Real-time extraction of neuromorphic features for nuclear and industrial security

51

robust and risk-aware ordering decisions. This structural feature explains its

superior performance, especially in data-limited or highly volatile demand

environments.

Overall, the results confirm that integrating probabilistic learning into

inventory control significantly enhances both economic efficiency and operational

reliability. The Bayesian framework provides a theoretically grounded and

practically effective alternative to traditional forecast-then-optimize approaches.

Future research could extend this work in several directions. First, multi-

item and multi-echelon inventory systems could be modeled to capture more

complex supply chain interactions. Second, hybrid approaches combining Bayesian

inference with deep learning methods may further improve predictive performance

while preserving uncertainty quantification. Finally, reinforcement learning

frameworks could be integrated with Bayesian updating to develop fully adaptive

inventory control policies for dynamic and non-stationary environments.

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