A machine learning-based branch and price algorithm for a sampled vehicle routing problem

Abstract

Planning of operations, such as routing of vehicles, is often performed repetitively in rea-world settings, either by humans or algorithms solving mathematical problems. While humans build experience over multiple executions of such planning tasks and are able to recognize common patterns in different problem instances, classical optimization algorithms solve every instance independently. Machine learning (ML) can be seen as a computational counterpart to the human ability to recognize patterns based on experience. We consider variants of the classical Vehicle Routing Problem with Time Windows and Capacitated Vehicle Routing Problem, which are based on the assumption that problem instances follow specific common patterns. For this problem, we propose a ML-based branch and price framework which explicitly utilizes those patterns. In this context, the ML models are used in two ways: (a) to predict the value of binary decision variables in the optimal solution and (b) to predict branching scores for fractional variables based on full strong branching. The prediction of decision variables is then integrated in a node selection policy, while a predicted branching score is used within a variable selection policy. These ML-based approaches for node and variable selection are integrated in a reliability-based branching algorithm that assesses their quality and allows for replacing ML approaches by other (classical) better performing approaches at the level of specific variables in each specific instance. Computational results show that our algorithms outperform benchmark branching strategies. Further, we demonstrate that our approach is robust with respect to small changes in instance sizes.

Application of Machine Learning in Supply Chain Management: A Comprehensive Overview of the Main Areas

“…Application of Machine Learning Algorithms in Transportation and Distribution. Vehicle routing problems (VRPs) are categorized as one of the most applicable issues in SCM [75][76][77][78][79][80]. Solving VRPs is one of the high and wellknown applications of ML in SCM.…”

Matheuristics and Column Generation for a Basic Technician Routing Problem

“…The authors of [15] provide a taxonomy of matheuristics for vehicle routing problems: Decomposition approaches are constructive heuristics solving iteratively sub-problems as in [48]; improvement heuristics apply local search heuristics with large MILP neighborhoods as in [49]; and CG-based heuristics derive primal heuristics from Lagrangian relaxations or CG and B&P algorithms as in [13,50]. The hybridization of ML techniques with CG algorithm was proven efficient for some vehicle routing problems [16,17].…”

“…In such case, all the variables related to the technician are zeros. Constraints (16) are the equivalent of previous flow constraints (4) for modeling routes for technician; the balance of flow is here considered regarding the ordinality k. Constraints ( 17)- (19) express time windows constraints, respectively, between two consecutive jobs and for starting and finishing time of technicians, similarly with constraints (6)- (8). Constraint (17) is the same constraint set as (6) by using relations (11).…”

“…CG approaches can be extended to CG heuristics or used in any hybrid heuristic to tackle large instances of vehicle routing problems [13][14][15]. Recently, the contribution of ML techniques for solving CG sub-problems is studied for some vehicle routing problems [16,17].…”

Enhancing Column Generation by a Machine-Learning-Based Pricing Heuristic for Graph Coloring

“…Machine Learning for combinatorial optimization has received a lot of attention in recent years (Bengio, Lodi, and Prouvost 2021). Existing studies have applied machine learning in a variety of ways, such as learning variable selection methods Gasse et al 2019;Liu et al 2020;Furian et al 2021) or node selection methods (He, III, and Eisner 2014;Furian et al 2021) for exact branch-and-bound solvers; learning to select the best algorithm among its alternatives based on the problem characteristics Khalil et al 2017b); learning to determine whether to perform problem reformulation (Kruber, Lübbecke, and Parmentier 2017;Bonami, Lodi, and Zarpellon 2018) or problem reduction Ding et al 2020); learning primal heuristics aiming to construct an optimal solution directly (Khalil et al 2017a;Kool, van Hoof, and Welling 2019); and learning to select columns for column generation (Morabit, Desaulniers, and Lodi 2021).…”