Uncertainty-Aware Delivery Delay Duration Prediction via Multi-Task Deep Learning

Stefan Faulkner; Reza Zandehshahvar; Vahid Eghbal Akhlaghi; Sebastien Ouellet; Carsten Jordan; Pascal Van Hentenryck

cs.LGFeb 23, 2026

Uncertainty-Aware Delivery Delay Duration Prediction via Multi-Task Deep Learning

Authors: Stefan Faulkner, Reza Zandehshahvar, Vahid Eghbal Akhlaghi, Sebastien Ouellet, Carsten Jordan, Pascal Van Hentenryck

Organizations: NSF Artificial Intelligence Institute for Advances in Optimization, H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA, USA · Kinaxis

Abstract

Accurate delivery delay prediction is critical for maintaining operational efficiency and customer satisfaction across modern supply chains. Yet the increasing complexity of logistics networks, spanning multimodal transportation, cross-country routing, and pronounced regional variability, makes this prediction task inherently challenging. This paper introduces a multi-task deep learning model for delivery delay duration prediction in the presence of significant imbalanced data, where delayed shipments are rare but operationally consequential. The model embeds high-dimensional shipment features with dedicated embedding layers for tabular data, and then uses a classification-then-regression strategy to predict the delivery delay duration for on-time and delayed shipments. Unlike sequential pipelines, this approach enables end-to-end training, improves the detection of delayed cases, and supports probabilistic forecasting for uncertainty-aware decision making. The proposed approach is evaluated on a large-scale real-world dataset from an industrial partner, comprising more than 10 million historical shipment records across four major source locations with distinct regional characteristics. The proposed model is compared with traditional machine learning methods. Experimental results show that the proposed method achieves a mean absolute error of 0.67-0.91 days for delayed-shipment predictions, outperforming single-step tree-based regression baselines by 41-64% and two-step classify-then-regress tree-based models by 15-35%. These gains demonstrate the effectiveness of the proposed model in operational delivery delay forecasting under highly imbalanced and heterogeneous conditions.

Figures & tables

Indices and Counts
Symbol	Domain/Type	Description
$N$	Positive integer	Number of shipments
$K$	Positive integer	Number of categorical features
$M$	Positive integer	Number of numerical features
$i$	$\{1,\ldots,N\}$	Shipment index
$k$	$\{1,\ldots,K\}$	Categorical feature index

Table 1: Mathematical notations and definitions.

Figure 1: Logistics network overview. Shipments $s_{i}\in\mathcal{S}$ flow from origin locations $L_{k}\in\mathcal{L}$ to destinations $R_{j}\in\mathcal{R}$ , characterized by physical features (weight, volume, items count) and timing information (planned departure/arrival).

Figure 2: Overview of the multi-task learning model. Input features are processed through dedicated embedding layers (categorical and numerical), concatenated, and passed to a shared MLP backbone that produces a hidden representation $\mathbf{h}_{i}$ . The backbone output feeds into three task-specific heads: a classification head for binary delay prediction, and dual regression heads for duration estimation. The classification probability $\hat{p}_{i}$ is concatenated with $\mathbf{h}_{i}$ and passed to both regression heads (dashed arrows). During training, samples are routed to regression heads based on ground truth labels; during inference, routing is determined by the classifier output ( $\hat{p}_{i}>0.5\rightarrow$ delayed head). Quantile predictions are calibrated via conformal prediction to produce prediction intervals.

Panel of Figure 3 — Figure 3: Delivery time distributions across locations showing distinct patterns of on-time (yellow) and delayed (blue) shipments. Main plots show the overall distribution, while insets focus on the delayed shipment patterns.

Figure 4: Distribution of shipment volumes across source locations.

Feature	Cardinality	Description
From Location ID	4	Source location identifier
From City	4	City of shipment origin
From Postal Code	4	Postal code of origin
From Country System ID	4	Country code for origin
To Location ID	198,019	Destination location identifier
To City	34,572	Destination city name

Table 2: Summary of Categorical Features

Feature	Unit	Description
Service Order Weight	Kilograms	Total weight of the shipment
Service Order Volume	Cubic meters	Total volume of the shipment
Total Number of Items	Count	Number of individual items in shipment
Destination City Latitude	Degrees	Destination city latitude coordinate
Destination City Longitude	Degrees	Destination city longitude coordinate
Distance	Kilometers	Direct distance between origin and destination

Table 3: Summary of Numerical Features

Delay Duration Estimation
Metric	Formula	Description
MAE (O/D)	$\displaystyle\frac{1}{\lvert\mathcal{D}_{\text{test}}^{}\rvert}\sum_{i\in\mathcal{D}_{\text{test}}^{}}\lvert y_{i}-\hat{y}_{i}^{(0.5)}\rvert$	Mean absolute error between actual delays and median predictions (days).
Avg-QL (O/D)	$\displaystyle\frac{1}{3\lvert\mathcal{D}_{\text{test}}^{}\rvert}\sum_{\alpha\in\{0.1,0.5,0.9\}}\sum_{i\in\mathcal{D}_{\text{test}}^{}}\rho_{\alpha}(y_{i}-\hat{y}_{i}^{(\alpha)})$	Average quantile loss across all three quantiles, where $\rho_{\alpha}(u)=\max(\alpha u,(\alpha-1)u)$ .
Uncertainty Quantification
Coverage (O/D)	$\displaystyle\frac{1}{\lvert\mathcal{D}_{\text{test}}^{}\rvert}\sum_{i\in\mathcal{D}_{\text{test}}^{}}\mathbb{I}[\hat{y}_{i}^{(0.1)}\leq y_{i}\leq\hat{y}_{i}^{(0.9)}]$	Proportion of true delays within prediction intervals. Target: 80% ( $\alpha=0.2$ ).
AIW (O/D)	$\displaystyle\frac{1}{\lvert\mathcal{D}_{\text{test}}^{}\rvert}\sum_{i\in\mathcal{D}_{\text{test}}^{}}(\hat{y}_{i}^{(0.9)}-\hat{y}_{i}^{(0.1)})$	Average interval width (days).

Table 4: Summary of Evaluation Metrics

Model	L1				L2				L3				L4
	Avg-QL $(\downarrow)$		MAE $(\downarrow)$		Avg-QL $(\downarrow)$		MAE $(\downarrow)$		Avg-QL $(\downarrow)$		MAE $(\downarrow)$		Avg-QL $(\downarrow)$		MAE $(\downarrow)$
	Ovr	Del	Ovr	Del	Ovr	Del	Ovr	Del	Ovr	Del	Ovr	Del	Ovr	Del	Ovr	Del
XGB-S1	0.05	0.52	0.10	1.43	0.13	0.58	0.33	1.48	0.17	0.59	0.41	1.85	0.16	0.64	0.32	1.70
CatB-S1	0.05	0.76	0.11	1.72	0.14	0.59	0.41	1.59	0.18	0.62	0.41	1.86	0.16	0.79	0.30	1.86
XGB-S2	0.11	0.46	0.23	1.02	0.20	0.52	0.40	1.16	0.28	0.40	0.60	0.90	0.20	0.67	0.41	1.40
CatB-S2	0.10	0.47	0.21	1.00	0.17	0.55	0.37	1.24	0.32	0.33	0.70	0.81	0.19	0.65	0.34	1.40

Table 5: Point prediction performance across locations. Best delayed performance per location highlighted in bold. Lower values indicate better performance. Ovr = overall instances, Del = delayed instances only.

Model	Type	L1			L2			L3			L4
		Cov $\uparrow$	AIW $\downarrow$	WS $\downarrow$	Cov $\uparrow$	AIW $\downarrow$	WS $\downarrow$	Cov $\uparrow$	AIW $\downarrow$	WS $\downarrow$	Cov $\uparrow$	AIW $\downarrow$	WS $\downarrow$
XGB-S1	Overall	0.843	0.381	0.926	0.769	0.650	2.135	0.823	0.663	3.128	0.700	0.438	3.131
XGB-S1	Delayed	0.435	1.442	8.411	0.422	1.406	9.914	0.455	1.346	8.453	0.197	0.862	10.768
CatB-S1	Overall	0.729	0.412	1.124	0.648	0.653	2.105	0.561	0.562	3.232	0.642	0.392	3.284
CatB-S1	Delayed	0.358	0.986	9.542	0.480	1.493	9.906	0.350	1.017	9.471	0.103	0.741	13.842
XGB-S2	Overall	0.864	0.184	2.255	0.769	0.438	3.983	0.647	0.719	5.481	0.768	0.153	4.095

Table 6: Uncertainty quantification performance (pre-calibration). Target coverage is 80% (

\alpha=0.2

). Best delayed performance per location highlighted in bold for Coverage and WS. For Coverage, higher is better; for AIW and WS, lower is better.

Model	Type	L1			L2			L3			L4
		Cov $\uparrow$	AIW $\downarrow$	WS $\downarrow$	Cov $\uparrow$	AIW $\downarrow$	WS $\downarrow$	Cov $\uparrow$	AIW $\downarrow$	WS $\downarrow$	Cov $\uparrow$	AIW $\downarrow$	WS $\downarrow$
XGB-S1	Overall	0.846	0.384	0.924	0.847	0.659	2.132	0.826	0.664	3.126	0.774	0.440	3.129
XGB-S1	Delayed	0.438	1.448	8.396	0.426	1.418	9.869	0.457	1.348	8.450	0.199	0.865	10.762
CatB-S1	Overall	0.812	0.578	1.386	0.764	0.816	2.348	0.678	0.694	2.986	0.728	0.524	3.086
CatB-S1	Delayed	0.362	0.992	9.538	0.484	1.502	9.873	0.406	1.024	9.436	0.108	0.749	13.718
XGB-S2	Overall	0.870	0.186	2.248	0.776	0.443	3.972	0.654	0.726	5.468	0.772	0.155	4.086

Table 7: Uncertainty quantification performance (post-calibration). Target coverage is 80% (

\alpha=0.2

). Best delayed performance per location highlighted in bold for Coverage and WS. For Coverage, higher is better; for AIW and WS, lower is better.

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Parameter	Range	Scale
n_estimators	[200, 2000]	linear
max_depth	[3, 12]	linear
learning_rate	[1e-3, 0.3]	log
subsample	[0.6, 1.0]	linear
colsample_bytree	[0.6, 1.0]	linear
gamma	[0, 5]	linear

Appendix

Table 8: Hyperparameter Search Space for XGBoost

Parameter	Range	Scale
iterations	[200, 1500]	linear
depth	[4, 12]	linear
learning_rate	[1e-3, 0.3]	log
l2_leaf_reg	[1, 30]	linear
border_count	[32, 255]	linear
bagging_temperature	[0, 10]	linear

Appendix

Table 9: Hyperparameter Search Space for CatBoost

Parameter	Range / Values
n_blocks	[2, 12]
d_hidden	{128, 192, 256, 320, 384}
dropout	[0.0, 0.5]
weight_decay	[1e-6, 3e-3]
gradient_clip_norm	{None, 1.0, 5.0}
batch_size	{256, 512, 1024, 2048}

Appendix

Table 10: Hyperparameter search space for the multitask deep learning model.

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The early detection of delayed cases in business processes is a critical capability for organizations. Predictive process monitoring (PPM) supports this task by using historical event logs to predict the remaining time of ongoing cases, enabling timely interventions to avoid missed deadlines and service level violations. Although remaining time prediction has advanced considerably through sophisticated deep learning architectures, little is known about the intrinsic difficulty of delay detection itself. Since performance is typically assessed using aggregate metrics, prior work provides limited insight into how models perform across the target distribution, especially on the operationally most critical cases with large delays. In this paper, we address this gap by analyzing the difficulty of delay detection. Across 14 event logs, we show that remaining times are typically strongly right-skewed, with only a small fraction of cases exhibiting large delays. Existing models capture the mode of this distribution well but perform poorly on high-delay cases. We further uncover pronounced heteroscedasticity, showing that predictive uncertainty increases with delay magnitude. Based on these findings, we evaluate approaches to mitigate the imbalance problem, but find only limited benefits, suggesting that the key underlying problem may not be imbalance but higher uncertainty associated with delayed cases. We show that this correlation can be exploited to substantially improve the identification of delayed cases. Overall, our work provides new insights into the sources of difficulty in delay detection and identifies uncertainty-aware modeling as a promising direction for future PPM research.

Abstract

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