Performances improvement of the column generation algorithm: application to vehicle routing problems

Column generation is a fundamental technique for solving large-scale combinatorial optimization problems such as unit commitment and vehicle routing, yet its performance is often limited by dual oscillation. This study explores the intrinsic cause of this phenomenon from the perspective of shadow price theory and demonstrates that dual oscillation arises from the lack of marginal interpretability of Lagrange multipliers when multiple dual solutions coexist. To address this issue, an improved column generation framework is proposed in which traditional multipliers are replaced with minimum-norm multipliers that possess clear economic meaning and act as directional shadow prices. A generalized pricing subproblem is formulated, and partial minimum-norm multipliers are obtained through convex quadratic optimization to guide column generation. Numerical experiments on a simplified single-period unit commitment case and large-scale cutting stock problems showed that the proposed approach eliminated invalid column generation and achieved speedy convergence to the optimal solution within only two iterations for the unit commitment case, and the classical column generation exhibited slow convergence with dual oscillation in large-scale scenarios while the improved algorithm achieved fast and stable convergence. The results indicate that the stabilization method enhances the consistency of dual variables and provides a more robust foundation for the theoretical and practical development of column generation algorithms.

In this paper, we introduce the Electric Vehicle Routing Problem with Flexible Time Windows (EVRPFTW) in which vehicles are allowed to serve customers before and after the earliest and latest time window bounds, respectively. The objective of this problem is to assign electric vehicles to feasible routes and make schedules with minimum total cost that includes the traveling costs, the costs of using electric vehicles and the penalty costs incurred for earliness and lateness. The proposed mathematical model is solved by a column generation procedure. To generate an integer solution, we solve an integer programming problem using the routes constructed by the column generation algorithm. We further develop a linear programming model to compute the optimal times to start service at each customer for the selected routes. A number of well-known benchmark instances is solved by our solution procedure to evaluate the operational gains obtained by employing flexible time windows.

An exact algorithm for a vehicle routing problem with time windows and multiple use of vehicles

In this paper, we address the electric vehicle routing problem with time windows and propose two branch-and-price-and-cut methods based on a column generation algorithm. One is an exact algorithm whereas the other is a heuristic method. The pricing sub-problem of the column generation method is solved using a label correcting algorithm. The algorithms are strengthened with the state-of-the-art acceleration techniques and a set of valid inequalities. The acceleration techniques include: (i) an intermediate column pool to prevent solving the pricing sub-problem at each iteration, (ii) a label correcting method employing the ng-route algorithm adopted to our problem, (iii) a bidirectional search mechanism in which both forward and backward labels are created, (iv) a procedure for dynamically eliminating arcs that connect customers to remote stations from the network during the path generation, (v) a bounding procedure providing early elimination of sub-optimal routes, and (vi) an integer programming model that generates upper bounds. Numerical experiments are conducted using a benchmark data set to compare the performances of the algorithms. The results favour the heuristic algorithm in terms of both the computational time and the number of instances solved. Moreover, the heuristic algorithm is shown to be specifically effective for larger instances. Both algorithms introduce a number of new solutions to the literature.

Column generation is an iterative method used to solve a variety of optimization problems. It decomposes the problem into two parts: a master problem and one or more pricing problems (PP). The total computing time taken by the method is divided between these two parts. In routing or scheduling applications, the problems are mostly defined on a network, and the PP is usually an NP-hard shortest path problem with resource constraints. In this work, we propose a new heuristic pricing algorithm based on machine learning. By taking advantage of the data collected during previous executions, the objective is to reduce the size of the network and accelerate the PP, keeping only the arcs that have a high chance to be part of the linear relaxation solution. The method has been applied to two specific problems: the vehicle and crew scheduling problem in public transit and the vehicle routing problem with time windows. Reductions in computational time of up to 40% can be obtained. Funding: The authors thank Giro, Inc., and the Natural Sciences and Engineering Research Council of Canada for financial support [Grant CRDPJ 520349-17].

Vehicle routing problem with fuzzy time windows

A column generation and Combinatorial Benders Decomposition algorithm for the Selective Dial-A-Ride-Problem

Routing problems are a class of combinatorial problems with many practical applications. Recently, end-to-end deep learning methods have been proposed to learn approximate solution heuristics for such problems. In contrast, classical dynamic programming (DP) algorithms guarantee optimal solutions, but scale badly with the problem size. We propose Deep Policy Dynamic Programming (DPDP), which aims to combine the strengths of learned neural heuristics with those of DP algorithms. DPDP prioritizes and restricts the DP state space using a policy derived from a deep neural network, which is trained to predict edges from example solutions. We evaluate our framework on the travelling salesman problem (TSP), the vehicle routing problem (VRP) and TSP with time windows (TSPTW) and show that the neural policy improves the performance of (restricted) DP algorithms, making them competitive to strong alternatives such as LKH, while also outperforming most other ‘neural approaches’ for solving TSPs, VRPs and TSPTWs with 100 nodes.KeywordsDynamic ProgrammingDeep LearningVehicle Routing

In response to the rapid increase in e-commerce, distribution services, and smart technology, businesses need to solve the vehicle routing problem (VRP) and its variants to improve service quality and cut operating costs. In this paper, we introduce the VRP with time windows and time costs (VRPTWTC), which is a new but practical VRP variant that adds time windows of customers and time costs of service constraints into the optimization of the traditional VRP problem. Furthermore, we propose a masking algorithm and combine it into our proposed end-to-end and data-driven deep reinforcement learning framework to solve the VRPTWTC. Experimental results show a shorter tour length and reasonable processing time than benchmarks both on a simulated dataset and a real enterprise-level dataset. Besides, violation of time window and total time cost, which is one of the most important criterions in our problem formulation, never occurred in our experiment under the real road connection conditions when using a real map API. To the best of our knowledge, our framework is the first to effectively and efficiently solve a VRP variant problem with practical time windows and time costs constraints with a real map API to calculate routes of VRP considering real road connection conditions on an enterprise-level dataset.

Models and hybrid methods for the onshore wells maintenance problem

A multiphase dynamic programming algorithm for the shortest path problem with resource constraints

We consider the multitasking scheduling problem on unrelated parallel machines to minimize the total weighted completion time. In this problem, each machine processes a set of jobs, while the processing of a selected job on a machine may be interrupted by other available jobs scheduled on the same machine but unfinished. To solve this problem, we propose an exact branch‐and‐price algorithm, where the master problem at each search node is solved by a novel column generation scheme, called in‐out column generation, to maintain the stability of the dual variables. We use a greedy heuristic to obtain a set of initial columns to start the in‐out column generation, and a hybrid strategy combining a genetic algorithm and an exact dynamic programming algorithm to solve the pricing subproblems approximately and exactly, respectively. Using randomly generated data, we conduct numerical studies to evaluate the performance of the proposed solution approach. We also examine the effects of multitasking on the scheduling outcomes, with which the decision maker can justify making investments to adopt or avoid multitasking.

A Vehicle Routing Problem with Flexible Time Windows

Consider a set of trips where each trip is specified a priori by a place of origin, a destination, a duration, a cost and a time interval within which the trip must begin. The problem is to minimize fleet size and total travel costs. This is a generalization of the m-travelling salesman problem. We compare results for three algorithms. The first one is an adaptation of the Carpaneto-Toth algorithm for the asymmetric travelling salesman problem: solution of nework problems by relaxing scheduling constraints, and branch-and-bound on flow variables. The second one consists in solution of network problems by relaxing scheduling constraints and branch-and-bound based on dividing the time windows. In the third algorithm, we use column generation on a set partitioning problem solved by simplex and branch-and-bound; columns are generated by a shortest path algorithm with time windows on the nodes.Key wordsRoutingSchedulingTime Windows

This paper proposes an abstraction of emerging vehicle routing problems, the Vehicle Routing Problem with Black Box Feasibility. In this problem the routes of a basic VRP need to satisfy an unknown set of constraints. A black box function to test the feasibility of a route is provided. This function is considered of non-linear complexity (in the length of the route). Practical examples of such problems are combinations of VRP with Loading problems or VRP with Scheduling problems. The difficulty in addressing the VRP with Black Box Feasibility lies in the unknown problem structure and the costly feasibility check. We propose a column generation-based approach to locally optimize this problem. Columns are heuristically generated by so-called Collector ants, executing a construction heuristic while guided by pheromones. To find an integer solution we solve an integer Set Partitioning Problem defined on the set of columns generated by the ants. We test the proposed approach on two applications from the literature, the Three-Dimensional Loading Capacitated Vehicle Routing Problem and the Multi-Pile Vehicle Routing Problem, showing the applicability of our approach and its good behavior compared to dedicated approaches.