Learning-Based Branch-and-Price Algorithms for the Vehicle Routing Problem with Time Windows and Two-Dimensional Loading Constraints

A capacitated vehicle routing problem with two-dimensional loading constraints is addressed. Associated with each customer are a set of rectangular items, the total weight of the items, and a time window. Designing exact algorithms for the problem is very challenging because the problem is a combination of two NP-hard problems. An exact branch-and-price algorithm and an approximate counterpart are proposed to solve the problem. We introduce an exact dominance rule and an approximate dominance rule. To cope with the difficulty brought by the loading constraints, a new column generation mechanism boosted by a supervised learning model is proposed. Extensive experiments demonstrate the superiority of integrating the learning model in terms of CPU time and calls of the feasibility checker. Moreover, the branch-and-price algorithms are able to significantly improve the solutions of the existing instances from literature and solve instances with up to 50 customers and 103 items. Summary of Contribution: We wish to submit an original research article entitled “Learning-based branch-and-price algorithms for a vehicle routing problem with time windows and two-dimensional loading constraints” for consideration by IJOC. We confirm that this work is original and has not been published elsewhere, nor is it currently under for publication elsewhere. In this paper, we report a study in which we develop two branch-and-price algorithms with a machine learning model injected to solve a vehicle routing problem integrated the two-dimensional packing. Due to the complexity brought by the integration, studies on exact algorithms in this field are very limited. Our study is important to the field, because we develop an effective method to significantly mitigate computational burden brought by the packing problem so that exactness turns to be achievable within reasonable time budget. The approach can be generalized to the three-dimensional case by simply replacing the packing algorithm. It can also be adapted for other VRPs when high-dimensional loading constraints are concerned. Broadly speaking, the study is a typical example of adopting supervised learning to achieve acceleration for operations research algorithms, which expands the envelop of computing and operations research. Hence, we believe this manuscript is appropriate for publication by IJOC.

Vehicle Routing with Stochastic Supply of Crowd Vehicles and Time Windows

The growth of e-commerce has increased demand for last-mile deliveries, increasing the level of congestion in the existing transportation infrastructure in urban areas. Crowdsourcing deliveries can provide the additional capacity needed to meet the growing demand in a cost-effective way. We introduce a setting where a crowd-shipping platform sells heterogeneous products of different sizes from a central depot. Items sold vary from groceries to electronics. Some items must be delivered within a time window, whereas others need a customer signature. Furthermore, customer presence is not guaranteed, and some deliveries may need to be returned to the depot. Delivery requests are fulfilled by a fleet of professional drivers and a pool of crowd drivers. We present a crowd-shipping platform that standardizes crowd drivers’ capacities and compensates them to return undelivered packages back to the depot. We formulate a two-stage stochastic model, and we propose a branch and price algorithm to solve the problem exactly and a column generation heuristic to solve larger problems quickly. We further develop an analytical method to calculate upper bounds on the supply of vehicles and an innovative cohesive pricing problem to generate columns for the pool of crowd drivers. Computational experiments are carried out on modified Solomon instances with a pool of 100 crowd vehicles. The branch and price algorithm is able to solve instances of up to 100 customers. We show that the value of the stochastic solution can be as high as 18% when compared with the solution obtained from a deterministic simplification of the model. Significant cost reductions of up to 28% are achieved by implementing crowd drivers with low compensations or higher capacities. Finally, we evaluate what happens when crowd drivers are given the autonomy to select routes based on rational and irrational behavior. There is no cost increase when crowd drivers are rational and select routes that have a higher compensation first. However, when crowd drivers are irrational and select routes randomly, the cost can increase up to 4.2% for some instances.

Models and Algorithms for the Bin-Packing Problem with Minimum Color Fragmentation

In the bin-packing problem with minimum color fragmentation (BPPMCF), we are given a fixed number of bins and a collection of items, each associated with a size and a color, and the goal is to avoid color fragmentation by packing items with the same color within as few bins as possible. This problem emerges in areas as diverse as surgical scheduling and group event seating. We present several optimization models for the BPPMCF, including baseline integer programming formulations, alternative integer programming formulations based on two recursive decomposition strategies that utilize decision diagrams, and a branch-and-price algorithm. Using the results from an extensive computational evaluation on synthetic instances, we train a decision tree model that predicts which algorithm should be chosen to solve a given instance of the problem based on a collection of derived features. Our insights are validated through experiments on the aforementioned applications on real-world data. Summary of Contribution: In this paper, we investigate a colored variant of the bin-packing problem. We present and evaluate several exact mixed-integer programming formulations to solve the problem, including models that explore recursive decomposition strategies based on decision diagrams and a set partitioning model that we solve using branch and price. Our results show that the computational performance of the algorithms depends on features of the input data, such as the average number of items per bin. Our algorithms and featured applications suggest that the problem is of practical relevance and that instances of reasonable size can be solved efficiently.

A Branch-and-Price Algorithm for Balancing Two-Sided Assembly Lines with Zoning Constraints

Two-sided assembly lines are widely used in the large-size product manufacturing industry, especially for automotive assembly production. Balancing the assembly line is significant for assembly process planning and assembly production. In this study, we develop a novel and exact method to optimize the two-sided assembly line balancing problem with zoning constraints (TALBz), in which the aim is to minimize the number of mated-stations considering the task restrictions. A mixed-integer programming model is employed to exactly describe the TALBz problem. To strengthen the computational efficiency, we apply Dantzig–Wolfe decomposition to reformulate the TALBz problem. We further propose a branch-and-price (B&P) algorithm that integrates the column generation approach into a branch-and-bound frame. Both the benchmark datasets with zoning constraints and without zoning constraints are tested to evaluate the performance of the B&P algorithm. The numerical results show that our proposed approach can obtain optimal solutions efficiently on most cases. In addition, experiments on the real-world datasets originating from passenger vehicle assembly lines are conducted. The proposed B&P algorithm shows its advantage in tackling practical problems with the task restrictions. This developed methodology therefore provides insight for solving large-scale TALBz problems in practice.