Probabilistic Heuristic for International Shipments

• The paper introduces a probabilistic heuristic that assigns international shipment flows to specific establishments using structured random sampling and calibrated parameters.
• It integrates empirical data from ports, commodities, and industrial sectors to closely match observed trade volumes and simulate urban freight dynamics.
• The model extends to hierarchical transit-time frameworks, optimizing dispatch scheduling through chance-constrained nonlinear optimization and gradient-based methods.

A probabilistic heuristic for international shipments refers to a behaviorally informed, transport-sensitive algorithm that assigns international freight flows to specific importer and exporter establishments within a domestic economy. Such frameworks integrate empirical data at the port, commodity, industry sector, and establishment levels, sampling candidate firms and shipment sizes according to rigorously defined probability distributions derived from trade flow, sectoral production/consumption shares, and operational constraints. Probabilistic dispatch models incorporating transit time uncertainty are also applied in hierarchical hub networks, including border crossings and customs processes, enabling systemic calibration against observed commodity movements and optimizing dispatch scheduling with respect to stochastic travel and clearance times. These approaches serve policy evaluation, network resilience analysis, and targeted infrastructure intervention, as demonstrated in agent-based metropolitan freight simulations and multimodal hierarchical routing studies (Ismael et al., 22 Nov 2025, Yousefzadeh et al., 2019).

1. Mathematical Foundations and Probability Distributions

The mathematical formulation underlying probabilistic heuristics for international shipments is based on structured random assignment that approximates observed port-level trade volumes and commodity flow patterns. Core sets and indices include:

• $\mathcal{J}$: international ports (land borders, seaports)
• $\mathcal{O}$: commodity groups (e.g., 15 SCTG)
• $\mathcal{I}$: NAICS industry sectors
• $\mathcal{Z}$: domestic zones (counties, regions)
• $E$: set of establishments

Parameters are established from disaggregated freight data (FAF) for annual tonnage $F_{j o}$ per port and commodity, and BEA Input-Output tables for sectoral production ($M^{prod}_{i o z}$) and consumption ($M^{cons}_{i o z}$) shares. The assignment process samples:

• Port for commodity $o$: $P[j|o] = F_{j o}/(\sum_{k} F_{k o})$
• Sector for zone $z$, commodity $o$: $$P[i|o,z] = M^{type}_{i o z}/(\sum_{h} M^{type}_{h o z})$$ where $type$ selects production (export) or consumption (import)
• Establishments: Uniform selection from size-filtered candidates $E_{i z}$
• Shipment tonnage: $d_e \sim \mathrm{Uniform}(lb, ub)$, subject to available port residual

Each shipment reduces residual port flow $R_{j o}$, with the implicit objective to minimize total discrepancy $E = \sum_{j,o} |R_{j o}|$ by driving all residuals to zero through probabilistic sampling (Ismael et al., 22 Nov 2025).

2. Input Data Sources, Model Structure, and Representation

Data integration is central to model fidelity. Key inputs include:

• FAF port-level flows ($F_{j o}$)
• Sector-commodity production/consumption shares (BEA Make-Use tables, mapped to SCTG via NAICS)
• Synthetic establishment attributes (via POLARIS freight-generation, including NAICS, location, employment, annual revenue)

Candidate pools of importer/exporter establishments are pre-filtered by employment or revenue threshold, and only eligible firms participate in assignment. Each international shipment is captured by specifying the supplier, receiver, commodity, and assigned annual tonnage. Export assignments link domestic establishment to port; imports link port to domestic establishment. Domestic drayage routing is post-processed through a multimodal router (POLARIS), although international leg distances (ocean miles) are excluded, reflecting the granularity of available shipment data (Ismael et al., 22 Nov 2025).

3. Parameter Calibration and Discrepancy Metrics

Trade volume bounds ($lb$, $ub$) are initialized to correspond to practical vehicle loads (e.g., one loaded semi per year to four per day). These parameters can be tuned against observed shipment-size histograms via Kullback-Leibler divergence minimization. Sector–commodity shares ($M^{prod}$, $M^{cons}$) directly reflect underlying IO tables and require no empirical fitting if taken as ground truth.

Discrepancies between modeled and observed flows are quantified exclusively through port × commodity residuals: $\sum_{j,o} |R_{j o}|$. While the existing scheme aims for exact matches by construction, extensions may utilize re-weighting or iterative proportional fitting to minimize squared assignment errors across ports, commodities, and sectors—a direction suggested for future calibration refinement (Ismael et al., 22 Nov 2025).

4. Algorithmic Implementation for Assignment and Scheduling

The algorithm proceeds in sequential steps:

1. Precompute sector–zone–commodity shares and eligible establishment sets.
2. Initialize residuals $R_{j o} \leftarrow F_{j o}$ for all ports and commodities.
3. For each trade type (export/import) and commodity:
   • While $R_{j o} > 0$ for any port: a. Draw port $j$ according to $F_{j o}$ b. Associate with relevant zone $z$ c. Sample sector $i$ according to $M^{type}_{i o z}$ d. Select establishment $e$ uniformly from $E_{i z}$ e. Sample $d \sim \mathrm{Uniform}(lb, ub)$, assign $\delta = \min(d, R_{j o})$ f. Create record $(supplier, receiver, o, \delta)$ g. Update $R_{j o} \leftarrow R_{j o} - \delta$
4. Repeat until assignment exhausts all observed flows.

This approach preserves observed aggregate flows and realistic establishment-level shipment sizes. Empirical deployment across four U.S. metropolitan regions (Atlanta, Chicago, Dallas-Fort Worth, Los Angeles) demonstrates exact matching to FAF port-level imports/exports, with regionally realistic counts of importers and exporters per port (Ismael et al., 22 Nov 2025).

5. Empirical Validation in Freight Networks

Applied in agent-based metropolitan freight simulations, the heuristic produces assignment statistics reflective of true network activity. For example, average internal importer counts per port are: Atlanta $\sim$39, Chicago $\sim$40, Dallas-Fort Worth $\sim$124, Los Angeles $\sim$417, while exporters per port range: Atlanta $\sim$48, Chicago $\sim$52, Dallas-Fort Worth $\sim$492, Los Angeles $\sim$38. Realistic counts are recovered due to size filtering in establishment pools. While the model does not explicitly generate international leg distance distributions, domestic drayage distances are subsequently assigned through detailed multimodal network routing (Ismael et al., 22 Nov 2025).

6. Hierarchical Probabilistic Transit-Time Models in Dispatch Decisions

Probabilistic heuristics in international shipment assignment extend naturally to hierarchical hub networks with uncertain transit and customs clearance times. Each shipment leg is modeled by a probability distribution:

• Truncated normal: $X_e \sim \mathcal{N}(\mu_e, \sigma_e^2)$, $f_e(t)$ on $[0,\infty)$
• Log-normal, gamma, or finite mixture distributions for empirical fit

Parameter estimation employs maximum likelihood methods (normal, log-normal, truncated normal, EM for mixtures). Joint, hierarchical, and correlated legs are accommodated via independence, hierarchical Bayesian, or multivariate normal frameworks as appropriate.

Dispatch scheduling is formulated as nonlinear optimization: minimize expected late-arrival, vehicle delay, and transfer costs over random transit and clearance times, subject to nonnegativity, capacity, and delivery window constraints. Chance-constrained sample-average approximations enforce required on-time delivery probabilities via Monte Carlo simulation. Algorithmic implementation utilizes homotopy continuation: transform an easy-to-solve problem into the target objective and follow the solution path using gradient-projection or LBFGS methods. Empirical convergence is achieved within 10–30 iterations, yielding 5–15% improvements over standard nonlinear optimizers in practical instances (Yousefzadeh et al., 2019).

7. Limitations and Proposed Extensions

Key limitations include:

• Unobserved establishment participation restricts assignments to "large" importers/exporters; uniform volume sampling may misrepresent true firm-scale heterogeneity.
• Sector–commodity shares are aggregated at the zoning level, omitting intra-zone variation.
• Port selection and establishment assignment are memoryless, lacking adaptation for port specialization.
• Correlations between firm revenue and shipment volume are not represented in current uniform sampling.

Proposed extensions and enhancements include:

• Replacement of uniform volume sampling with truncated log-normal or empirical distributions fitted to observed shipment or bill-of-lading data.
• Adoption of multinomial logit models for establishment selection, incorporating utility terms for port distance, firm size, and historical trade intensity.
• Iterative proportional fitting to globally minimize assignment discrepancies.
• Integration of cost-based port selection, e.g., incorporating drayage costs via $P[j] \propto F_{j o} \cdot \exp(-\theta C_{e,j})$.
• Joint optimization of international and domestic legs via large-scale decomposition.
• Hierarchical network adaptation to model customs clearance probabilistically, including sample-average approximations and fixed-threshold policies to manage chain-level delivery probability.

These methodologies facilitate rigorous modeling of international shipment assignments and dispatch scheduling under realistic behavioral and operational constraints in complex multimodal freight systems (Ismael et al., 22 Nov 2025, Yousefzadeh et al., 2019).