Branch-Price-and-Cut Accelerated with a Pricing for Integrality Heuristic for the Electrical Vehicle Routing Problem with Time Windows and Charging Time Slots

Branch-price-and-cut is the state-of-the-art exact method for solving many types of vehicle routing problems, and is particularly effective for vehicle routing problems with time windows. A well-known challenge in branch-price-and-cut is that the generation of columns is guided by information from the linear relaxation of the master problem, with no guarantee that they will be useful from an integer perspective. As a consequence, high-quality primal solutions are often found only after significant cutting and branching or the use of primal heuristics. In this work, based on the ideas of pricing for integrality, we propose a new primal heuristic for vehicle routing problems. The heuristic is designed to generate columns that are more likely to be part of high-quality integer solutions. It begins by constructing a partial integer solution from a given column pool and then iteratively searches for columns that complement this solution. The search is done by modifying the pricing problem with respect to the partial solution, linear program dual information as well as previously generated columns in the heuristic. Computational tests are performed on the electrical vehicle routing problem with time windows extended with charging time slots, a problem that has both scheduling and routing aspects, making it well-suited to evaluate the performance of the proposed heuristic. The results show that the proposed heuristic closes 30% - 40% of the root node gap on average in comparison to a restricted master heuristic.

💡 Research Summary

The paper addresses a critical shortcoming of the Branch‑Price‑and‑Cut (BPC) framework when applied to vehicle routing problems, namely that column generation is driven solely by reduced‑cost information from the linear relaxation of the master problem. Consequently, columns that are useful for high‑quality integer solutions often appear late in the search, delaying strong primal bounds. To remedy this, the authors introduce a “pricing for integrality” primal heuristic specifically adapted to the electric vehicle routing problem with time windows and charging time slots (EVRPTW‑CTS).

The heuristic follows a destroy‑and‑repair paradigm. In the destroy phase, several partial integer solutions are constructed from the existing column pool by removing a subset of columns. The removal is performed either randomly or via a spatially aware destroy mechanism that targets geographically close customers, thereby preserving the spatial structure of the problem while creating meaningful gaps (unserved customers, unscheduled charging slots). In the repair phase, the algorithm iteratively generates new columns that complement the partial solution. This is achieved by solving a modified pricing problem whose objective blends the usual dual values from the LP relaxation with additional penalty/reward terms: (i) a reward for covering rows (customers) that remain uncovered, (ii) a penalty for exceeding the capacity of charging time slots, and (iii) dynamic penalties that discourage duplication of already generated columns. The modified objective explicitly steers the pricing sub‑problem toward columns that are likely to appear in an optimal integer solution.

A key technical innovation lies in exploiting the labeling algorithm traditionally used for the ESPPRC (Elementary Shortest Path Problem with Resource Constraints) in vehicle routing. Rather than generating a single column per pricing iteration, the authors let the labeling routine produce several promising columns at once. Because the labeling algorithm already explores a large set of partial paths, this multi‑column generation incurs negligible extra computational effort while dramatically increasing the diversity of the column pool. The heuristic also incorporates a spatial destroy mechanism, which improves the relevance of the generated columns for the EVRPTW‑CTS’s combined routing‑scheduling nature.

The computational study focuses on the EVRPTW‑CTS, an extension of the classic EVRPTW that introduces capacitated charging resources available only during specific time slots. This addition creates a hybrid set‑covering and set‑packing structure: each customer must be served, and each charging slot can be used by a limited number of vehicles. The problem reflects realistic constraints in heavy‑duty electric truck fleets, where shared charging infrastructure must be scheduled to avoid bottlenecks.

Implementation is built on the generic GCG (General Column Generation) framework, part of the SCIP Optimization Suite. The authors augment GCG with problem‑specific components: an accelerated labeling algorithm, a suite of cutting planes, tailored branching rules, a restricted master heuristic, and the proposed pricing‑for‑integrality heuristic. All code and benchmark instances are released publicly.

Experimental results show that the new heuristic closes 30 %–40 % of the root‑node optimality gap on average compared with the standard restricted‑master heuristic. The improvement is most pronounced when the column pool is still sparse, indicating that the heuristic effectively injects high‑quality columns early in the BPC process. Moreover, the multi‑column generation strategy yields a richer set of routes without increasing runtime, and the spatial destroy mechanism helps align generated routes with the limited charging‑slot schedule.

In summary, the contributions are threefold: (1) a novel primal heuristic for BPC that integrates integrality considerations directly into the pricing sub‑problem, (2) the definition and public release of the EVRPTW‑CTS, a realistic and challenging benchmark that combines routing, time‑window, and charging‑slot constraints, and (3) empirical evidence that the heuristic substantially improves early primal bounds, thereby reducing the size of the branch‑and‑bound tree for large‑scale electric vehicle routing problems. The work opens avenues for further research on integrality‑aware column generation in other combinatorial optimization contexts.