PSR-GIP Heuristic for Vehicle Routing

• PSR-GIP Heuristic is a multi-stage meta-heuristic designed to solve complex, multi-modal vehicle routing challenges in disaster management.
• It builds hierarchical decision trees and uses preferential selection combined with route fragment integration to uphold compatibility and causality constraints.
• The method optimizes emergency logistics by producing near real-time, scalable routes where traditional exact methods like MILP become computationally prohibitive.

The PSR-GIP Heuristic is a multi-stage, problem-specific meta-heuristic designed for high-quality solution generation in complex, multi-modal vehicle routing problems, particularly in the context of disaster management support systems. "PSR-GIP" refers to the process of Preferential Selection of Routes (PSR) from node-variant-specific decision trees, combined with Generation, Integration, and Perturbation (GIP) of route elements. The methodology systematically utilizes hierarchical compatibility constraints and sequential causality tracking, rendering it effective for large-scale, highly-constrained routing problems in settings where classical exact methods such as MILP are computationally prohibitive (Banerjee et al., 16 Sep 2025).

1. Core Principles and Objective

The central objective of the PSR-GIP Heuristic is the minimization of overall emergency operation time, formalized as either the minimum makespan ($z_x$), i.e., the maximum route duration among all vehicles, or the total sum of route durations ($z_s$). The approach addresses a “rich” vehicle routing paradigm that incorporates:

• Heterogeneous vehicle fleets with multi-trip capacity
• Distributed depots across geographically diverse locations
• Multiple transport modes and transshipment ports
• Stringent compatibility constraints at both the vehicle–cargo and transshipment–cargo level
• Temporal and causal ordering for multimodal transshipments and deliveries

This structure aligns the heuristic for integration into GIS-based decision support systems, enabling near real-time feedback and robust solution delivery under operationally realistic constraints.

2. Decision Tree Construction and Preference Encoding

The algorithm commences with the construction of a domain-specific, hierarchical decision tree per problem node. The node structure is as follows:

Level in Tree	Encoded Object	Functionality
Trunk	Supply/demand node	Target service point (e.g., delivery/pickup, transshipment)
Branches	Vehicle types	Compatibility with mode, availability from depots
Twigs	Vertices	Reachable end-points or transit points
Leaves	Cargo types (with hyperparams)	Feasible cargo assignments, capacity and compatibility

Leaves include a multiplier ($m$) and exponent (%%%%3%%%%) as hyper-parameters, supporting the calculation of a preference score, guiding the preferential selection (the “PSR” phase) among potential SREs (Smallest Route Elements). This representation inherently prunes infeasible options (e.g., incompatible cargo/vehicle/vertex combinations), serving as a natural constraint enforcement and preference propagation mechanism.

3. Smallest Route Element (SRE) Generation and Causality Dict Tracking

On the basis of the preference-weighted decision tree, the heuristic generates SREs by decomposing the overall routing task into the fundamental logistical moves, each satisfying crucial compatibility and causality properties. For each SRE:

• SET 1: Vertices as sources (typically warehouses) for delivery cargo
• SET 2: Central node (core demand)—can represent simultaneous or split-servicing nodes
• SET 3: Vertices absorbing pickup cargo (typically relief centers)

Each SRE is constructed such that the cumulative vehicle load code (VLC) sums to zero, preserving mass-balance across the movement chain. For SREs involving transshipment, temporal dependencies are meticulously tracked using a causality dictionary (CD), which records order-sensitive transshipment chains and waiting times.

4. Integration and Logical Assembly into Route Clusters

Integration of SREs into full vehicle routes is managed through a modular, multi-logic system:

• SREs are first molded into route portions (RoPrs) by ordering elements of SET 1 → SET 2 → SET 3
• Multiple permutations are allowed for SETs 1 and 3 while SET 2 remains ordered centrally
• Route portions are incorporated into route clusters (RoCu) representing complete routes per vehicle using various logics:
  • Random selection
  • MinTime insertion (choosing the lowest incremental time)
  • Vertex similarity (placing new RoPrs to exploit previous visits)
  • Additional allocation and matching schemes, including randomized shuffling across cascades, to diversify the solution space

If a vehicle’s route contains a partial transshipment trail, new SREs are integrated only after complying with causality as per the CD, maintaining correct delivery precedence and waiting time calculation.

5. Perturbation and Local Improvement

Once initial route clusters are established, the heuristic executes a perturbation phase to escape local minima and enhance solution quality:

• Vertex positions within each route cluster are randomly selected for potential shifting
• Moves are validated for feasibility (especially SRE order preservation and compatibility)
• Upon each shift, solution metrics—such as vehicle load state, total duration, and waiting times—are recalculated
• If the perturbation yields improved makespan or route sum objectives, it is retained; otherwise, the modification is undone

This perturbation mechanism, combined with repeated cascaded application of integration logic, results in substantial improvement over initial greedy or combinatorial solutions, effectively navigating the highly constrained and multimodal search space.

6. Performance, Scalability, and Comparative Assessment

The heuristic’s efficacy is demonstrated through computational studies on both moderate and large instances:

• For small-scale problems where MILP is tractable, the gap between the best PSR-GIP solution and MILP optima is only a few percent in key metrics (makespan $z_x$, total route duration $z_s$)
• For large-scale, integer-heavy problems where MILP cannot produce a feasible solution within reasonable time frames, the PSR-GIP heuristic consistently produces high-quality routes within minutes
• Evaluation metrics include relative gap measures ($$\text{Improvement} (\%) = \frac{S_I - S_F}{S_F} \times 100$$), focusing on both the best route duration and aggregate durations across vehicles

The algorithm’s design maximizes parallel solution generation (across multiple cascades and integration logics), supporting rapid deployment in operational settings where decision speed is paramount.

7. Relevance to Disaster Management Decision Support

Designed explicitly for disaster operations, the PSR-GIP heuristic delivers practical benefits:

• Outputs are immediately usable in GIS-based platforms, detailing complete vehicle itineraries, mode transitions, cargo transfer chains, and transshipment waiting times
• Explicit causality and compatibility tracking ensures valid, executable plans under diverse, time-critical, and dynamically changing conditions
• The modular, cascaded structure permits solution refinement, partial updating, or full re-execution as new data arrives or operational circumstances shift

This suggests that PSR-GIP is well-suited to the requirements of emergency response, logistics planning for humanitarian relief, or any domain demanding fast generation of large-scale, highly-constrained multimodal route networks when exact optimization is infeasible.

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In summary, the PSR-GIP Heuristic embodies a sophisticated approach to rich vehicle route planning under severe operational constraints, leveraging preferential decision-tree construction, compatibility-aware fragment assembly, causality tracking, and localized perturbation to achieve solution quality and computational performance unattainable by classical MILP in large-scale, real-world applications (Banerjee et al., 16 Sep 2025).