Supply chain ordering must balance profit, customer service and upstream order variability (the bullwhip effect). This study evaluates whether a two-stage pipeline–a Liquid Neural Network (LNN) encoder with an adaptive-leak recurrent cell followed by an XGBoost readout–offers a useful operational trade-off for multi-tier ordering, rather than assuming hybrid superiority.
An LNN encoder represents temporal demand dynamics and XGBoost regresses its frozen latent states; the encoder is explicitly distinguished from canonical Liquid Time-Constant and closed-form continuous-time networks. Under a leakage-free protocol, hyperparameters are selected on a separate development seed and frozen before testing on untouched days from ten unseen demand seeds. A matched linear-readout ablation, paired statistics with multiplicity correction, an observed-order DataCo replay, cost sensitivity analysis and an end-to-end latency benchmark are used.
Transformer ($1.738M), XGBoost ($1.737M) and LNN+XGBoost ($1.737M) attain nearly identical mean system profit, with no Holm-adjusted difference between the hybrid and either baseline. Relative to the exact-matched linear readout, LNN+XGBoost gains $14,477 (95% CI $4,811–$24,143; d_z = 1.07), although the family-wise Holm result (p = 0.072) does not indicate a statistically significant difference. Its manufacturer demand-referenced variance ratio is 2.61 versus 2.44 (XGBoost) and 2.48 (Transformer), with a fill rate of 0.996. The Q-network minimizes variance (1.43) but sacrifices profit and service, the DataCo replay favors standalone XGBoost and the hybrid is not latency-optimal.
The study provides the first leakage-free, ablation-controlled evaluation of an adaptive-leak-LNN–XGBoost ordering pipeline and characterizes a profit–service–bullwhip trade-off with explicit operating conditions rather than claiming universal superiority.
1. Introduction
Supply chain ordering must manage fluctuating demand, inventory imbalance and upstream variance amplification. Long short-term memory (LSTM) and reinforcement learning can model dynamics but impose training and tuning costs; XGBoost is efficient on tabular features but does not natively encode sequences. This motivates testing whether a compact temporal encoder followed by tree regression offers a useful operational trade-off.
Liquid time-constant (LTC) and closed-form continuous-time networks use input-dependent state dynamics inspired by biological nervous systems (Hasani et al., 2020, 2022). Their hidden states respond adaptively to incoming sequences after training; this state adaptation does not imply online updating of model weights. The present encoder borrows the adaptive-state and time-constant ideas but is presented as an adaptive-leak LNN variant rather than a reproduction of the canonical LTC or CfC equations. We therefore call the overall encoder a liquid neural network (LNN) and disclose its adaptive-leak cell as the implementation used in this study.
This study evaluates a two-stage LNN + XGBoost model that sequentially composes liquid-inspired dynamic feature extraction with tree regression in a cascaded pipeline, not a jointly optimized architecture. The research question is: how does a frozen adaptive-state encoder followed by an XGBoost readout affect profit, service and upstream order amplification under an independently tested multi-tier ordering simulation? The latency benchmark in Section 4.4 tests, rather than assumes, whether a compact encoder translates into faster inference.
The study makes three bounded contributions. First, it tests a previously unreported adaptive-leak-LNN–XGBoost ordering pipeline and uses a matched linear-readout ablation to identify the incremental value of the tree stage rather than attributing performance to unspecified “hybrid synergy.” Second, it evaluates profit, service and both chained and demand-referenced bullwhip measures on an untouched test period across ten unseen demand seeds, with effect sizes and multiplicity correction. Third, it combines a multi-tier simulation with an observed-order replay, economic sensitivity analysis and end-to-end latency measurement, thereby identifying both the pipeline's useful operating region and the conditions under which simpler models are preferable.
2. Related work and research gap
This section reviews machine learning and deep learning techniques for supply chain forecasting and ordering optimization – continuous-time and liquid-inspired recurrent models, LSTM, transformers, XGBoost and hybrid models – and identifies the gap addressed by the proposed pipeline.
2.1 Continuous-time and liquid-inspired recurrent models
LTC networks model neuron dynamics using liquid time constants so that hidden states respond continuously to sequential inputs (Hasani et al., 2020). Hasani et al. (2022) subsequently derived closed-form continuous-time networks and reported speedups for those specific architectures and tasks. These findings motivate compact adaptive-state encoders, but do not justify labeling every leaky recurrent cell as a canonical LTC/CfC implementation or assuming lower end-to-end latency. We accordingly distinguish the simplified LNN used here from LTC/CfC and measure the complete GPU–CPU pipeline directly.
To our knowledge, LNN encoders with adaptive-leak cells followed by tree readouts have not been evaluated for multi-tier ordering with explicit profit, service and bullwhip outcomes. This bounded empirical gap motivates the present study; it is not a claim that the simplified encoder is a canonical LTC/CfC implementation.
2.2 LSTM and hybrid models in supply chain forecasting
LSTM captures long-term dependence (Graves, 2013; Benidis et al., 2022), while recent supply-chain hybrids combine clustering, recurrent networks, convolution or attention (Ji et al., 2024; Cao et al., 2024; Jahin et al., 2024). These studies primarily improve forecasts; they provide less evidence on closed-loop multi-tier ordering, bullwhip measures and end-to-end deployment cost.
2.3 Transformer models in supply chain forecasting
Transformer models use self-attention to model long-sequence dependence, although simpler time-series baselines can remain competitive (Zeng et al., 2022). Wibawa et al. (2022) illustrated a complementary convolutional approach, and Aguiar-Pérez and Pérez-Juárez (2023) reviewed deep demand forecasting in another operational domain. This study directly measures whether a compact LNN encoder plus XGBoost offers a useful outcome–efficiency trade-off.
2.4 XGBoost in supply chain forecasting
XGBoost is accurate and efficient on many tabular prediction tasks (Chen and Guestrin, 2016). Its feature-based regression is a natural second stage after temporal encoding, while standalone XGBoost supplies an essential simpler baseline. Whether the encoder adds value is therefore tested by both standalone XGBoost and a matched linear readout rather than assumed from model descriptions.
2.5 Deep reinforcement learning in supply chain optimization
Deep reinforcement learning has been applied to multi-echelon ordering, including discrete-action deep Q-network (DQN) policies for the beer game (Oroojlooyjadid et al., 2017). The present reward-trained Q-network comparator does not reproduce that canonical discrete-action design and is labeled accordingly.
2.6 Related studies on ordering optimization strategies
Ordering optimization typically integrates demand forecasting, safety stock and cost management. Chaharsooghi et al. (2008) used reinforcement learning for beer-game ordering; Silver et al. (1998) established the classical safety-stock and demand-forecast foundation for multi-tier ordering; and Bertsimas and Thiele (2006) developed robust optimization under demand uncertainty. These approaches adjust orders from forecasted demand to optimize profit but do not incorporate an LNN or XGBoost.
2.7 Hybrid models and research gaps
Convolutional and hybrid forecasting studies demonstrate the value of learned temporal representations (Zhao and Wang, 2017; Borovykh et al., 2017), and modern optimization frameworks enable systematic tuning (Akiba et al., 2019). To our knowledge, no prior study has evaluated the specific LNN–XGBoost cascade used here for multi-tier ordering. The unresolved question is not merely predictive accuracy, but whether this cascade improves profit and service without worsening bullwhip or deployment cost on data not used for model selection.
3. Methodology
This section follows the audited data flow in Figure 1: demand generation, propagation, feature engineering, inventory-target modeling, order selection, development-only hyperparameter tuning and untouched-test evaluation.
The six-stage flowchart shows the leakage-free experimental and evaluation workflow. Controlled synthetic inputs feed a development-only phase using seed 32, with days 1–657 used to fit candidate models and days 658–876 used for ten Optuna T P E trials per model. One configuration per model is then frozen, with no seed-specific test tuning. Final evaluation uses ten unseen seeds (42–51), with days 1–876 used for fitting and days 877–1,095 reserved for untouched testing. In the causal ordering stage, L N N states are frozen before XGBoost, the matched ridge readout reuses the same encoder, and current orders arrive after one day. Independent outcomes include system and echelon profit, demand-referenced and chained variance ratios, fill rate, service volatility, paired tests, Holm correction, and effect sizes. Final-test outcomes never return to Optuna or alter the frozen configuration.Overall pipeline: input generation, two-stage LNN → XGBoost modeling, development-only Optuna tuning, frozen-model order simulation and untouched-test evaluation
The six-stage flowchart shows the leakage-free experimental and evaluation workflow. Controlled synthetic inputs feed a development-only phase using seed 32, with days 1–657 used to fit candidate models and days 658–876 used for ten Optuna T P E trials per model. One configuration per model is then frozen, with no seed-specific test tuning. Final evaluation uses ten unseen seeds (42–51), with days 1–876 used for fitting and days 877–1,095 reserved for untouched testing. In the causal ordering stage, L N N states are frozen before XGBoost, the matched ridge readout reuses the same encoder, and current orders arrive after one day. Independent outcomes include system and echelon profit, demand-referenced and chained variance ratios, fill rate, service volatility, paired tests, Holm correction, and effect sizes. Final-test outcomes never return to Optuna or alter the frozen configuration.Overall pipeline: input generation, two-stage LNN → XGBoost modeling, development-only Optuna tuning, frozen-model order simulation and untouched-test evaluation
3.1 Consumer demand generation
Consumer demand combines a baseline, seasonal and weekly cycles, and Gaussian noise:
Demand is constrained to be nonnegative and generated under fixed seeds; Section 4.2 gives the numerical specification.
3.2 Demand propagation across layers
The four layers are consumers, retailers, distributors and manufacturers. Each operational echelon receives the immediately downstream order:
This cascade permits direct measurement of incremental and system-wide order amplification.
3.3 Feature engineering
Features comprise current and lagged orders, inventory, sales, recent volatility, seasonality and normalized time. They are Min–Max normalized and processed in ten-day windows.
3.4 Forecasting and order decision process
3.4.1 Inventory-target models
The learned models estimate a one-day-ahead inventory target under the causal reference dynamics. This horizon matches the fixed one-day replenishment lead time. The compared models are:
Liquid neural network + XGBoost (LNN + XGBoost): for input xt, activation at, learned base leak λ0 and time constant τ, the implemented update is
Here dt = 0.1, Wrec is initialized with 50% sparsity and τ is learned. This is a discrete, liquid-inspired adaptive-leak cell, an LNN variant rather than the canonical LTC or CfC equation. The encoder is trained first; its states are frozen and flattened, and XGBoost then predicts the inventory target. There is no feedback from XGBoost to the encoder.
XGBoost, LSTM and transformer: tabular, recurrent and attention-based baselines.
Q-network comparator: a reward-based neural baseline with replay and exploratory action proposals; it is not labeled as canonical DQN.
3.4.2 Safety stock calculation
Safety stock is dynamically computed to mitigate demand variability:
Where SSbase is the buffer, SSfactor,i is layer-specific and is recent demand volatility.
3.4.3 Causal inventory flow and order selection
At the start of day t, the order placed at t − 1 arrives. Available inventory is Ai(t) = Ii(t − 1) + Oi(t − 1), realized sales are Si(t) = min{Ai(t), Qi(t)} and ending inventory is Ii(t) = Ai(t) − Si(t), where Q1(t) = D(t) and Qi(t) = Oi−1(t) for i > 1. Thus, a newly placed order does not arrive on the same day.
All policies use the same operational order batch of 16 units; this is distinct from the neural training mini-batch. Nonclassical policies evaluate batch-feasible candidates between a safety-stock-adjusted lower bound and an upper bound derived from 1.5 times the recent mean incoming order. Each candidate is valued one lead-time step ahead using the predicted inventory target and the same revenue, purchase, holding and shortage accounting:
The selected nonnegative order is rounded upward to the common operational batch:
The classical comparator uses an echelon order-up-to rule under the same lead time and order batch.
3.5 Hyperparameter optimization
Hyperparameters are selected once on a separate development demand seed (32). Within that seed, days 1–657 fit candidate models and days 658–876 tune them through ten Optuna TPE trials per model. The objective is manufacturer cumulative profit on that development-only tuning period:
The resulting configuration is frozen before final evaluation. For each unseen evaluation seed (42–51), the model is fitted on days 1–876 and evaluated only on untouched days 877–1,095. No final-test outcome enters hyperparameter selection. Tuned parameters include architecture size, learning rate, training mini-batch, epochs and safety-stock baseline; the operational order batch remains fixed at 16 for every policy.
3.6 Feature correlation analysis
For descriptive transparency, pairwise Pearson correlations are computed among the six displayed input features selected for each model family and echelon and visualized in Appendix 3. Coefficients range from −1 to +1, and the unit diagonal is each variable's self-correlation. This is an association analysis rather than SHapley Additive exPlanations (SHAP) or feature attribution: it does not measure a feature's contribution to a prediction, establish predictive importance or support causal conclusions.
3.7 Performance evaluation
The system's effectiveness is evaluated using an efficiency metric comparing actual to theoretical maximum profit:
Where:
Assumes perfect demand fulfillment without holding or shortage costs. A 7-day moving average smooths daily fluctuations:
Additional metrics include:
Inventory turnover: Sales divided by average inventory.
Service level: Proportion of demand fulfilled.
Shortage cost: Costs from unmet demand.
Holding cost: Expenses from inventory storage.
Order volatility: Standard deviation of orders.
Inventory-target MAE: Mean absolute error of the one-day inventory-target prediction.
Because mitigating the bullwhip effect is a central objective, we use two complementary measures. For seed r, the system-wide variance ratio compares manufacturer orders with end-consumer demand: VRr = Var(O3,r)/Var(Dr); values above 1 indicate amplification, and lower is better. We report the mean and standard deviation of VRr over ten seeds. The per-layer, chained bullwhip effect measure (BEM) isolates the amplification introduced at each echelon:
So that BEMi measures how much layer i amplifies the order variance it receives from layer i − 1 (with layer 1 measured against consumer demand). The chained ratios are distinct from the system-wide layer-3-to-consumer ratio. For each seed, peak amplification is max(O3)/max(D). At layer i, fill rate is ∑tSi(t)/∑tQi(t), where Si is realized sales and Q1 = D, Qi = Oi−1 for i > 1; the reported fill rate is the mean across layers. Service volatility is analogously the cross-layer mean of the standard deviation of Si(t)/Qi(t). All summary values are computed directly from saved validation tensors.
Because every model uses the same ten unseen demand seeds and untouched test days, inference preserves pairing. Friedman tests assess omnibus differences; Kendall's W = χ2/[N(k − 1)] quantifies rank concordance. All pairwise profit comparisons use paired t-tests with Holm family-wise-error correction. For the targeted matched hybrid-versus-linear comparison, we additionally report , a 95% confidence interval, and an exact Wilcoxon signed-rank test as a distribution-robust check.
4. Experimental research and analysis
4.1 Experiment purpose
The purpose is to evaluate six learning configurations: LNN + XGBoost, its matched linear-readout ablation, standalone XGBoost, LSTM, transformer and a Q-network decision-learning baseline. A classical echelon order-up-to policy provides a nonlearning benchmark. Performance is evaluated across retailer, distributor and manufacturer layers on data excluded from tuning.
4.2 Experiment method
The supply chain is structured with four layers:
Layer 0: Consumers
Layer 1: Retailers
Layer 2: Distributors
Layer 3: Manufacturers
Each demand realization contains 1,095 days. Model selection and final evaluation are separated at both seed and time levels:
Development seed 32: days 1–657 for fitting candidates and days 658–876 for 10-trial Optuna tuning.
Unseen evaluation seeds 42–51: days 1–876 for fitting the frozen configuration and days 877–1,095 for final testing.
The ten evaluation seeds and their final 219-day periods are never used by Optuna. Hyperparameters are common across all evaluation seeds, eliminating seed-specific best-of-ten selection and preserving an untouched paired test.
Initial conditions include:
Initial inventory for each layer: 100 units
Order fulfillment lead time: Fixed at 1 day
The cost and price structure is as follows:
Unit costs: [0, 30, 45, 60] for layers 0–3, respectively
Unit prices: [0, 70, 100, 130] for layers 0–3, respectively
Holding cost rate: 0.03 per unit per day
Shortage cost rate: 0.03 per unit per day
Consumer demand at layer 0 combines deterministic cycles and random noise as a controlled synthetic process:
Where t is the time step, and is Gaussian noise. Demand is constrained to be nonnegative, and fixed random seeds (42–51 for 10 runs) ensure reproducibility.
The experiment compares six learning configurations:
Feature engineering involves constructing a 10-dimensional feature vector per layer, including:
Current demand, lagged orders (t − 1, t − 2), lagged inventory (t − 1, t − 2), lagged sales (t − 1)
Volatility indicators: Standard deviation of orders and demand over the past 5 days
Seasonal indicator:
Normalized time:
The scaler is fitted only on the fitting period and applied unchanged to tuning or test observations. Features use a 10-day sliding window. Test-time attention normalization over the full sequence is not used, preventing future observations from influencing current representations.
The training process is as follows:
Models estimate the next-day inventory target, matching the one-day lead time.
For LNN + XGBoost, the encoder is trained first using MSE loss, frozen and followed by XGBoost on flattened latent states. The matched linear ablation reuses the exact frozen encoder and ordering configuration.
The Q-network comparator is trained with a profit-plus-service reward and replay.
Optuna performs ten trials per model only on development seed 32. The objective is tuning-period manufacturer profit. The single selected configuration is frozen across evaluation seeds. Training mini-batches of 4 or 8 affect optimization only; every policy faces the same 16-unit operational order batch.
During untouched testing (days 877–1,095), orders are selected causally each day:
Inventory-target prediction: Models estimate the next-day reference inventory target; the order signal is exponentially smoothed with α = 0.3.
Safety stock calculation:
Where σD is the standard deviation of demand over the past 10 days.
Order range exploration: Candidate orders use the common 16-unit batch and are bounded by the safety-stock-adjusted signal and 1.5 times the recent mean incoming order. Candidates are evaluated one lead-time step ahead; a new order never enters same-day inventory.
Profit calculation:
The order that maximizes profit is selected and adjusted to the nearest multiple of 16 units to satisfy batch constraints (see Table 1).
Comparison of six learning configurations
| Model | Components | Configuration details |
|---|---|---|
| LNN + XGBoost | LNN encoder (adaptive-leak cell) | 32–128 units (step 32), Xavier initialization, adaptive leak, learned τ, AdamW (learning rate 10–5 to 10–3), 10–20 epochs |
| XGBoost | Regressor on flattened frozen states, 50–150 trees, maximum depth 3–5, learning rate 0.01–0.3 | |
| LNN linear-readout ablation | Frozen LNN + ridge readout | Reuses the exact evaluation-seed encoder weights, state window, frozen hyperparameters and ordering parameters; only XGBoost is replaced by ridge regression (α = 1) |
| Standalone XGBoost | XGBoost | Trained on engineered features, 50–150 trees, maximum depth 3–5, learning rate 0.01–0.3 |
| LSTM | Long Short-Term Memory | 64–256 hidden units (step 64), 1–3 layers, one-day inventory target, Adam optimizer, gradient clipping (maximum norm 0.5) |
| Transformer | Transformer | Model dimensions 64–256, 2–8 attention heads, 1–3 layers, multi-head attention (dropout rate 0.1), gradient clipping (maximum norm 0.1) |
| Q-network baseline | Reward-trained neural network | 64–256 hidden units, replay buffer 20,000, exploratory action proposals, and profit-plus-service reward; treated as a Q-network-style comparator rather than a canonical discrete-action DQN |
4.3 Experiment results
Results below use only untouched days 877–1,095 of evaluation seeds 42–51. Hyperparameters were frozen before these demand realizations were evaluated. Table 5 reports system-wide profit – defined here as the sum of the three-echelon operating margins under fixed layer prices and costs, computed identically for every model (see the definition following Table 5) – while Figure 2 and Tables 2 and 3 report demand-referenced and chained order amplification (see Tables 4 and 5).
The grouped bar chart reports mean demand-referenced variance ratios Var(Oi)/Var(D), with standard-deviation error bars across ten unseen test seeds, for the retailer, distributor, and manufacturer echelons. Seven model bars are shown at each echelon: L N N+XGBoost, L N N linear, XGBoost, L S T M, Transformer, Q-network, and Order-up-to. A dashed horizontal line at 1 indicates no amplification. Retailer ratios range from 1.40 to 1.58. Distributor ratios range from 1.36 for the Q-network to 2.25 for Order-up-to. Manufacturer ratios range from 1.43 for the Q-network to 3.21 for Order-up-to; the other manufacturer values are 2.44 for XGBoost, 2.48 for Transformer, 2.57 for L N N linear, 2.61 for L N N+XGBoost, and 2.67 for L S T M. Lower values indicate less order-variance amplification relative to consumer demand.Untouched-test demand-referenced variance ratio Var(Oi)/Var(D) by echelon (mean ± SD across 10 unseen seeds)
The grouped bar chart reports mean demand-referenced variance ratios Var(Oi)/Var(D), with standard-deviation error bars across ten unseen test seeds, for the retailer, distributor, and manufacturer echelons. Seven model bars are shown at each echelon: L N N+XGBoost, L N N linear, XGBoost, L S T M, Transformer, Q-network, and Order-up-to. A dashed horizontal line at 1 indicates no amplification. Retailer ratios range from 1.40 to 1.58. Distributor ratios range from 1.36 for the Q-network to 2.25 for Order-up-to. Manufacturer ratios range from 1.43 for the Q-network to 3.21 for Order-up-to; the other manufacturer values are 2.44 for XGBoost, 2.48 for Transformer, 2.57 for L N N linear, 2.61 for L N N+XGBoost, and 2.67 for L S T M. Lower values indicate less order-variance amplification relative to consumer demand.Untouched-test demand-referenced variance ratio Var(Oi)/Var(D) by echelon (mean ± SD across 10 unseen seeds)
Demand-referenced bullwhip ratio Var(Oi)/Var(D) (mean ± SD over ten untouched test seeds). Lower is better; values above one indicate amplification relative to consumer demand
| Model | Layer 1 | Layer 2 | Layer 3 |
|---|---|---|---|
| LNN + XGBoost | 1.47 ± 0.05 | 2.00 ± 0.08 | 2.61 ± 0.19 |
| LNN linear | 1.40 ± 0.05 | 1.88 ± 0.07 | 2.57 ± 0.13 |
| XGBoost | 1.43 ± 0.05 | 1.83 ± 0.10 | 2.44 ± 0.15 |
| LSTM | 1.48 ± 0.03 | 1.99 ± 0.06 | 2.67 ± 0.14 |
| Transformer | 1.46 ± 0.03 | 1.92 ± 0.09 | 2.48 ± 0.14 |
| Q-network | 1.47 ± 0.04 | 1.36 ± 0.09 | 1.43 ± 0.15 |
| Order-up-to | 1.58 ± 0.05 | 2.25 ± 0.14 | 3.21 ± 0.27 |
Chained BEM Var(Oi)/Var(Oi−1) (mean ± SD). Layer 1 uses consumer demand as its denominator
| Model | Layer 1 | Layer 2 | Layer 3 |
|---|---|---|---|
| LNN + XGBoost | 1.47 ± 0.05 | 1.36 ± 0.04 | 1.31 ± 0.06 |
| LNN linear | 1.40 ± 0.05 | 1.34 ± 0.03 | 1.36 ± 0.05 |
| XGBoost | 1.43 ± 0.05 | 1.28 ± 0.04 | 1.33 ± 0.02 |
| LSTM | 1.48 ± 0.03 | 1.34 ± 0.03 | 1.34 ± 0.04 |
| Transformer | 1.46 ± 0.03 | 1.32 ± 0.05 | 1.29 ± 0.03 |
| Q-network | 1.47 ± 0.04 | 0.92 ± 0.05 | 1.05 ± 0.07 |
| Order-up-to | 1.58 ± 0.05 | 1.42 ± 0.05 | 1.42 ± 0.09 |
System-wide bullwhip and service metrics (mean ± standard deviation over 10 seeds). Variance ratio and peak amplification compare layer-3 manufacturer orders with consumer demand. Fill rate and service volatility are first computed per layer and then averaged across layers
| Model | Variance ratio | Peak amplification | Fill rate | Service volatility |
|---|---|---|---|---|
| LNN + XGBoost | 2.612 ± 0.186 | 1.263 ± 0.066 | 0.996 ± 0.001 | 0.037 ± 0.003 |
| LNN linear | 2.568 ± 0.131 | 1.426 ± 0.126 | 0.991 ± 0.005 | 0.055 ± 0.011 |
| XGBoost | 2.438 ± 0.153 | 1.223 ± 0.029 | 0.995 ± 0.001 | 0.044 ± 0.004 |
| LSTM | 2.673 ± 0.136 | 1.345 ± 0.105 | 0.997 ± 0.001 | 0.037 ± 0.005 |
| Transformer | 2.480 ± 0.137 | 1.223 ± 0.029 | 0.996 ± 0.002 | 0.037 ± 0.006 |
| Q-network | 1.430 ± 0.149 | 0.816 ± 0.019 | 0.953 ± 0.004 | 0.092 ± 0.002 |
| Order-up-to | 3.207 ± 0.270 | 1.552 ± 0.390 | 0.990 ± 0.002 | 0.056 ± 0.007 |
Untouched-test system-wide cumulative profit (mean ± 95% CI, N = 10 unseen seeds) and one-day inventory-target MAE (mean ± SD). System-wide profit is a stylized aggregate echelon margin (see text), reported identically for every model
| Model | Mean cumulative profit (USD) | Inventory-target MAE |
|---|---|---|
| Transformer | 1, 737, 797 ± 4, 471 | 11.57 ± 0.85 |
| XGBoost | 1, 737, 362 ± 4, 352 | 16.95 ± 0.44 |
| LNN + XGBoost | 1, 737, 106 ± 3, 121 | 14.79 ± 0.40 |
| LSTM | 1, 735, 030 ± 3, 701 | 19.14 ± 0.96 |
| LNN linear | 1, 722, 629 ± 10, 832 | 21.53 ± 2.57 |
| Order-up-to | 1, 719, 828 ± 7, 906 | 24.45 ± 0.87 |
| Q-network | 1, 617, 572 ± 7, 236 | 7.11 ± 0.39 |
The Q-network has the lowest manufacturer-to-consumer variance ratio (1.43) but also the lowest profit and fill rate, illustrating that variance damping alone is not sufficient. Among the high-profit models, XGBoost has the lowest manufacturer ratio (2.44), followed by transformer (2.48), LNN + XGBoost (2.61) and LSTM (2.67). The classical order-up-to policy has the highest ratio (3.21). LNN + XGBoost improves fill rate and service stability relative to its matched linear readout, but its variance ratio is slightly higher (2.61 versus 2.57).
Transformer has the highest mean test profit ($1.738 M), followed closely by XGBoost and LNN + XGBoost. Their confidence intervals overlap, and Holm-adjusted paired comparisons show no difference between LNN + XGBoost and XGBoost or transformer (both p = 1.000). The matched readout contrast favors XGBoost by $14,477 (95% CI $4,811-$24,143; dz = 1.07; paired t9 = 3.39, raw p = 0.008; exact Wilcoxon p = 0.0039), but its Holm-adjusted p = 0.072 does not cross the family-wise 0.05 threshold. Thus the targeted effect is sizable and directionally consistent, but not multiplicity-robust in this ten-seed test.
Here system-wide profit is the sum of the retailer, distributor and manufacturer echelon operating margins (revenue less purchase, holding and shortage costs) under fixed layer prices and costs. Because each echelon's purchase cost is charged at its own layer cost rather than netted against the downstream layer's sale price, this is a stylized aggregate echelon profit for cross-model comparison, not a consolidated entity profit net of internal transfers; all models are scored identically, so the comparison is consistent. The final column reports the one-day inventory-target mean absolute error (MAE), the prediction accuracy underlying each policy.
Notably, the inventory-target MAE does not track profit rank: the Q-network has the lowest MAE (7.11) yet the lowest profit, while transformer, XGBoost and LNN + XGBoost span 11.6–17.0 MAE at nearly identical profit. This confirms that prediction accuracy alone does not determine ordering-policy value, motivating the profit, service and bullwhip outcomes as the primary criteria.
Raw outcomes, not a model-set-dependent composite, are used for inference. Friedman tests across the six learning configurations reject equal ranks for all four primary outcomes (Table 6). The classical policy remains a descriptive benchmark outside these six-model omnibus tests.
Friedman repeated-measures tests on paired untouched-test outcomes across six learning configurations
| Outcome | χ2(5) | p | Kendall's W |
|---|---|---|---|
| System profit | 33.60 | 2.86 × 10–6 | 0.672 |
| Variance ratio | 33.60 | 2.86 × 10–6 | 0.672 |
| Fill rate | 37.21 | 5.45 × 10–7 | 0.744 |
| Service volatility | 37.54 | 4.66 × 10–7 | 0.751 |
These omnibus effects primarily separate the Q-network's low-variance/low-service profile from the high-profit models. They do not establish superiority among transformer, XGBoost, LNN + XGBoost and LSTM, whose mean profits differ by less than 0.2%.
The Pearson heatmaps in Appendix 3 show that the direction and magnitude of pairwise feature associations vary across model families and echelons. Several order, demand, sales and inventory variables are moderately or strongly correlated, while some displayed inputs are weakly related. These patterns may reflect shared temporal structure and engineered lags, but they are not interpreted as model explanations or causal mechanisms.
4.4 Runtime and computational cost
We benchmark the actual frozen configurations selected by the development procedure, using a batch of 32 ten-step sequences, ten warm-up passes, 100 timed repetitions and Compute Unified Device Architecture (CUDA) synchronization on an RTX 4060 Laptop GPU and Intel i7-13700H (Table 7). Both LNN pipelines include GPU-to-CPU state transfer and their CPU readout. The 128-unit LNN has 18,049 neural parameters, far fewer than the selected transformer (1.78 M), but LNN + XGBoost is the slowest measured pipeline at 20.26 ± 2.75 ms per batch. Parameter count therefore does not imply end-to-end speed.
Actual frozen-configuration inference benchmark (batch 32, sequence length 10; mean ± SD over 100 repetitions after ten warm-ups)
| Model | Neural parameters/config | Batch latency (ms) | Samples/s |
|---|---|---|---|
| LNN + XGBoost | 18,049 + 150 trees | 20.260 ± 2.753 | 1,579 |
| LNN linear | 18,049 + ridge | 13.282 ± 2.972 | 2,409 |
| XGBoost | 150 trees, depth 4 | 7.016 ± 0.685 | 4,561 |
| LSTM | 274,689 | 1.511 ± 0.499 | 21,184 |
| Transformer | 1,780,865 | 7.053 ± 3.548 | 4,537 |
| Q-network | 10,689 | 0.198 ± 0.083 | 161,980 |
4.5 Proof-of-concept replay using observed DataCo order quantities
As an observed-order replay, we use the public DataCo Smart Supply Chain dataset (Constante et al., 2019), comprising 180,519 order-line records whose order dates span 2015–01 through 2018–01. Order Item Quantity is aggregated to a daily series by calendar day, giving approximately 1,127 consecutive daily buckets; missing days are retained as zero observed quantity, and values above the 99th percentile of the daily series are winsorized to that threshold to limit the influence of extreme spikes. These transaction-derived quantities are a demand proxy: they may reflect realized orders or sales and need not equal unconstrained latent demand. The first 60% of the chronological series (the earliest calendar days) is used for fitting and the final 40% for replay over ten model-initialization seeds (42–51). Price, cost, holding penalty, shortage penalty, lead time and batch size are experimental assumptions rather than observed operating records. Because this single-series replay has one demand stream rather than the multi-echelon cascade of the primary experiment, we evaluate the deployment-relevant policies for that setting – the classical base-stock benchmark, standalone XGBoost, LSTM, standalone LNN and the LNN + XGBoost hybrid – which retrain directly on the single-echelon trace without structural change. We omit the transformer, whose multi-head attention configuration was tuned for the multi-tier sequence structure and would require separate re-optimization on this trace, and the Q-network, whose reward and action design is defined over the multi-echelon simulation rather than a fixed observed-order series. The exercise tests a policy against an empirical temporal trace, not against a complete real multi-echelon inventory history.
Table 8 reports mean ± standard deviation. All policies are loss-making under the assumed cost configuration, so the comparison concerns loss mitigation. The deterministic base-stock and XGBoost policies repeat exactly and therefore have zero initialization variance. The hybrid reduces the base-stock loss by 35.9% and raises fill rate from 0.602 to 0.622, but standalone XGBoost has the smallest mean loss and highest fill rate, while LSTM has the lowest variance ratio. The replay demonstrates feasibility under an observed temporal trace, not external validation of a complete industrial supply chain and not superiority.
Closed-loop DataCo comparison (mean ± standard deviation over 10 initialization seeds). All policies are loss-making under the assumed cost configuration; less-negative profit is better
| Policy | Total profit ($) | Fill rate | Bullwhip ratio |
|---|---|---|---|
| Base-stock | −1, 245, 695 ± 0 | 0.602 ± 0.000 | 0.480 ± 0.000 |
| XGBoost-Opt | −708, 205 ± 0 | 0.626 ± 0.000 | 0.513 ± 0.000 |
| LSTM-Opt | −784, 491 ± 64, 816 | 0.622 ± 0.003 | 0.353 ± 0.040 |
| LNN-Opt | −786, 868 ± 300, 645 | 0.621 ± 0.012 | 0.445 ± 0.063 |
| Hybrid-Opt (LNN + XGBoost) | −797, 921 ± 36, 242 | 0.622 ± 0.001 | 0.532 ± 0.056 |
Because the DataCo ordering rules do not take monetary costs as inputs, we conduct a transparent ex-post accounting sensitivity analysis on the same saved trajectories rather than imply policy re-optimization. Table 9 varies shortage penalty from 0.5 to 2.0 times price and doubles holding cost. The hybrid improves on base-stock under every scenario, but standalone XGBoost remains the most profitable policy. Thus, the feasibility result is not an artifact of one cost setting, while the conclusion that the hybrid is not universally preferable is also stable (see Table 9).
DataCo cost sensitivity: mean total profit in USD millions over ten initialization seeds. Policy trajectories are held fixed; only their accounting valuation changes
| Scenario | Base | XGB | LSTM | LNN | Hybrid |
|---|---|---|---|---|---|
| Low shortage penalty | 2.097 | 2.436 | 2.390 | 2.399 | 2.378 |
| Baseline costs | −1.246 | −0.708 | −0.784 | −0.787 | −0.798 |
| High shortage penalty | −7.932 | −6.996 | −7.134 | −7.158 | −7.150 |
| High holding cost | −1.252 | −0.773 | −0.834 | −0.813 | −0.855 |
5. Discussion and recommendations
On untouched test periods, transformer, XGBoost and LNN + XGBoost are practically tied in mean profit, with overlapping intervals and no Holm-adjusted pairwise differences. LNN + XGBoost has a high fill rate but more manufacturer variance amplification than XGBoost and transformer. The matched linear contrast yields a large standardized profit difference and an exact Wilcoxon result in the same direction, but the p = 0.072 Holm result prevents a family-wise significance claim. The Q-network provides the strongest variance damping only by accepting materially lower profit and fill rate. DataCo cost scenarios favor standalone XGBoost, and the selected LNN + XGBoost pipeline is slowest in the measured inference benchmark. The defensible conclusion is competitive performance with explicit trade-offs, not dominance or proven nonlinear-readout superiority.
5.1 Managerial implications
The results support only a conditional, locally re-estimated deployment rule. If low latency and tabular inputs dominate, standalone XGBoost is the simpler candidate. The LNN pipeline is a candidate only when a local matched-ablation test shows material value and the organization accepts additional pipeline overhead. If variance amplification dominates, lower-variance baselines should be examined. These are screening recommendations, not universal prescriptions; deployment requires local lead times, capacities, costs and prospective validation.
5.2 Limitations
Several limitations temper these findings. First, the primary experiment uses synthetic demand, a fixed one-day lead time and no supplier capacity constraints. DataCo contributes an observed order-quantity trace but not actual inventory states, lost demand, capacities, costs or lead times; its sensitivity is ex-post accounting on fixed trajectories. Second, the implemented LNN uses a simplified adaptive-leak recurrent cell rather than a canonical LTC/CfC implementation, and the cascade is not jointly optimized. Third, Optuna targets manufacturer profit on one separate development seed, which protects the final test but may not span all regimes. Fourth, ten evaluation seeds limit power for small effects, as illustrated by the ablation's raw and multiplicity-adjusted results. Fifth, runtime covers one hardware configuration. Future work should use multiple industrial datasets, stochastic lead times, capacity constraints, rolling-origin external validation, cost-aware retraining and canonical continuous-time comparators.
5.3 Data and code availability
The DataCo source is identified by DOI 10.17632/8gx2fvg2k6.5. The synthetic generator, development seed (32), evaluation seeds (42–51), chronological split, frozen hyperparameters, per-seed test arrays and analysis outputs are included in the reproducibility package accompanying the revision. The audited environment used Python 3.12.10, PyTorch 2.6.0+cu124, XGBoost 3.1.2, Optuna 4.5.0, scikit-learn 1.6.1, pandas 2.3.3 and NumPy 2.2.6. Inference hardware and timing protocol are reported in Section 4.4.
6. Conclusion
This study evaluates a two-stage LNN + XGBoost pipeline under a leakage-free development/test design. Across 10 unseen demand seeds, Transformer, XGBoost and LNN + XGBoost achieve nearly identical mean test profit. The hybrid improves mean profit by $14,477 over its exact-matched linear readout with a large standardized effect, but that contrast does not remain significant after correction across all profit pairs. XGBoost and transformer also produce lower manufacturer variance ratios, while the Q-network trades substantial profit and service for stronger variance damping. DataCo replay and every accounting scenario favor standalone XGBoost, and the actual frozen hybrid is not latency-optimal. LNN + XGBoost is therefore a competitive but more complex option whose value must be established locally; it is not a universal replacement for simpler models.
The supplementary material for this article can be found online
