History Routing: Data-Driven Route Optimization

• History Routing is a data-driven routing paradigm that leverages historical decisions and GPS trajectories to inform route selection in logistics and networks.
• It employs methods like Markovian route learning and trajectory-based routing, blending statistical transition matrices with spatial grid indexing for real-world relevance.
• The approach improves accuracy and robustness by integrating smoothing techniques, adaptive fallbacks, and risk-aware dynamic programming under uncertainty.

History routing refers to a class of routing paradigms in which historical data—comprising past decisions, observed trajectories, or empirical stochastic costs—informs and directs the computation of routes in logistical, vehicular, or networked systems. Rather than relying solely on explicit graph models, hand-crafted objective functions, or deterministic edge costs, history routing leverages statistical regularities in recorded or observed behavior, planner preference, or realized arc costs to generate solutions that better reflect implicit objectives and real-world variability.

1. Definitions and Foundational Concepts

History routing departs from classical shortest-path or vehicle routing models by grounding route computation in empirical data. Two principal approaches are observed:

• Markovian Route Learning: Historical route solutions are treated as samples from a first-order Markov process, with the system learning a probabilistic transition matrix that captures the likelihood that a node (or stop) $i$ is followed by node $j$ in real-world routings (Canoy et al., 2019).
• Trajectory-Based Routing: Instead of operating over a road graph, the system indexes and follows real GPS trajectory points. Route search proceeds by traversing the spatio-temporal “cloud” of trajectories, optionally falling back on succinct graph-based representations only where historical data is sparse (Siampou et al., 2 Nov 2024).

Both classes of methods contrast sharply with conventional distance- or time-based optimization, which typically minimize a predetermined cost function under explicit constraints, irrespective of actual historical routing choices or realized travel data.

2. History-Based Vehicle Routing via Markov Models

In the Markovian framework for history routing, the solution space consists of n historical VRP solutions $H = \langle h_1, ..., h_n \rangle$ over a universe of stops $\Sigma = \{s_0, ..., s_t\}$. The probability of traversing an arc $(i, j)$ is learned as a transition $$t_{ij} = Pr(\text{next stop} = s_j \mid \text{current} = s_i)$$, with the Markov assumption that only the current position influences choice of next stop. Historical solutions are represented as binary adjacency matrices $A^k$, aggregated via weighted summation:

$F = \sum_{k=1}^n w_k A^k$

where $w_k$ reflects uniformity, recency, or instance similarity (e.g., Jaccard similarity between client sets). To avoid overfitting to historical sparsity, Laplace smoothing is applied:

$$t_{ij} = \frac{f_{ij} + \alpha}{N_i + \alpha \cdot d}$$

where $f_{ij}$ is the arc frequency, $N_i$ is the total outgoing count from $i$, $d = |\Sigma|$, and $\alpha \geq 0$ is the smoothing parameter.

The learned transition matrix $T$ transforms the VRP into a probabilistic “most-likely sequence” problem. To solve for the best routing $X$, arc costs in the classical VRP model are substituted as $c'_{ij} = -\log t_{ij}$. The resulting optimization is

$\min \sum_{i \neq j} c'_{ij} x_{ij}$

subject to standard CVRP constraints, enabling the use of existing solvers unchanged (Canoy et al., 2019).

3. Trajectory-Based History Routing

A distinct paradigm, trajectory-based (or history-based) routing uses time-ordered empirical trajectories $T = \langle (p_1, ts_1), …, (p_n, ts_n) \rangle$, where each $p_k$ is a GPS coordinate and $ts_k$ a timestamp (Siampou et al., 2 Nov 2024). Instead of map-matching or per-edge travel time estimation, the method overlays a spatial grid and indexes trajectory (and optionally, road network) points in per-cell B-trees.

Route computation from source $s$ to destination $d$ interleaves two major search mechanisms:

• Trajectory Following: Expands only through consecutive or cell-neighboring trajectory points.
• Road Fallback: When trajectory coverage is low, switches to road network segment traversal, penalty-weighted.

The search employs a cost function combining trajectory and (optionally) road transition costs, controlled by parameters:

• Road penalty $\alpha \geq 0$ penalizes transitions onto the road network.
• Continuity reward $rw \geq 0$ rewards remaining on a single trajectory.

Blending between trajectory and road-based expansions is adaptively tuned as a function of observed spatial coverage, with the cost equation

$$\text{Cost}(p_k, p_m) = \alpha C_\text{traj}(p_k, p_m) + (1-\alpha) C_\text{road}(p_k, p_m)$$

where $\alpha$ grows with trajectory coverage $c$ (Siampou et al., 2 Nov 2024).

4. Robust History-Dependent Routing under Uncertainty

History-dependent routing in stochastic environments introduces dynamic programming formulations that exploit the available statistical information about arc costs. Let $G = (V, A)$ with random arc costs $C_a$, objective budget $T$, and risk function $f$, the baseline Bellman recursion for the value function $u_i(t)$ is

$$u_i(t) = \max_{j \in V(i)} \int_0^{\bar a} p_{ij}(\omega) u_j(t - \omega) d\omega$$

where routing policy can depend on full past history (“history-dependent policy”) (Flajolet et al., 2014). In practice, arc cost probabilities $p_{ij}$ may be unknown but constrained via ambiguity sets (e.g., confidence intervals for the mean, mean absolute deviation, or quantiles), enabling a robust min-max formulation:

$$\sup_\pi \inf_{p_a \in P_a\ \forall a} \mathbb{E}_p [f(T - X_\pi)]$$

Efficient solutions exist via discretized label-setting algorithms and inner linear programs, providing performance guarantees that depend on regularity properties of $f$ and the discretization step $\Delta t$ (Flajolet et al., 2014).

5. Comparative Evaluation and Empirical Findings

Empirically, history routing can achieve high fidelity to actual planner or driver behavior and offers substantial computational and maintenance efficiencies:

Approach	Data/Model	Key Strengths	Limitation Observed
Markovian routing	Historical route plans	High accuracy, seamless with CVRP solvers, adapts to drift	VRP scope, integral constraints
Trajectory-based	Raw GPS trajectories	Captures real-world behavior/traffic, low maintenance	Sparse coverage/cold-start
Robust DP	Probe-driven stochastic	Risk-aware planning, explicit confidence modeling	Model complexity

• In real-world vehicle routing, a history-based Markov model reproduced human planner preferences with lower route and arc difference than standard distance-based VRP solutions, solved ~6000x faster, and was robust to changes in client sets given basic smoothing (Canoy et al., 2019).
• Trajectory-based routing achieved travel time MAE as low as 0.8 minutes and spatial MAE of 310 meters with dense trajectory coverage, outperforming or matching commercial map-based APIs as coverage improved. Higher $\alpha$ and continuity reward further reduced travel time errors by up to 0.3 minutes (Siampou et al., 2 Nov 2024).
• For robust adaptive routing, history-dependent dynamic programming with ambiguity sets outperformed empirical and deterministic LET approaches in both average and worst-case on-time arrival probabilities, particularly with limited sample sizes (Flajolet et al., 2014).

6. Robustness, Limitations, and Open Directions

History routing offers natural robustness to unmodeled or changing operational preferences. In Markovian settings, Laplace smoothing and similarity-based weighting make the method resilient to previously unseen arcs or client-set drift. Trajectory-based routing’s adaptive fallback to road networks mitigates sparse-data scenarios, though cold-start and inability to immediately reflect real-world changes (e.g., new roads or closures) remain practical challenges (Canoy et al., 2019, Siampou et al., 2 Nov 2024).

In stochastic or risk-aware settings, ambiguity set design and tractable robust optimization with distributional uncertainty offer principled worst-case protection but may carry additional computational cost (Flajolet et al., 2014).

Open research avenues include richer models (higher-order Markov, sequence learning), driver- or vehicle-specific transition matrices, interactive planner feedback, and hybrid history–model-based routing frameworks that learn to blend or adapt structural parameters online (Canoy et al., 2019, Siampou et al., 2 Nov 2024). Other promising directions are region- and time-adaptive tuning of blending parameters, and incorporation of real-time probe or IoT trajectory feeds.

7. Historical Context and Broader Relevance

History routing shares conceptual ancestry with empirical-data-driven approaches in broader routing algorithm evolution. The shift from hand-coded distance-vector and link-state algorithms in network routing (Sinha et al., 17 Mar 2024) towards hybrid or learning-based schemes parallels the adoption of history-based, data-driven route planning in logistics and transportation. In both domains, leveraging observed behaviors and outcomes—rather than exclusively model-driven optimization—has enabled systems to more accurately capture real operational preferences, adapt to evolving environments, and scale with reduced maintenance overhead. The continuing co-evolution of history-based and model-based techniques, including reinforcement learning and robust optimization, remains foundational to advancements in both vehicle and network routing.