Louis-Martin Rousseau

Antoine François, Quentin Cappart, Louis-Martin Rousseau

Combinatorial optimization is the field devoted to the study and practice of algorithms that solve NP-hard problems. As Machine Learning (ML) and deep learning have popularized, several research groups have started to use ML to solve combinatorial optimization problems, such as the well-known Travelling Salesman Problem (TSP). Based on deep (reinforcement) learning, new models and architecture for the TSP have been successively developed and have gained increasing performances. At the time of writing, state-of-the-art models provide solutions to TSP instances of 100 cities that are roughly 1.33% away from optimal solutions. However, despite these apparently positive results, the performances remain far from those that can be achieved using a specialized search procedure. In this paper, we address the limitations of ML approaches for solving the TSP and investigate two fundamental questions: (1) how can we measure the level of accuracy of the pure ML component of such methods; and (2) what is the impact of a search procedure plugged inside a ML model on the performances? To answer these questions, we propose a new metric, ratio of optimal decisions (ROD), based on a fair comparison with a parametrized oracle, mimicking a ML model with a controlled accuracy. All the experiments are carried out on four state-of-the-art ML approaches dedicated to solve the TSP. Finally, we made ROD open-source in order to ease future research in the field.

Augustin Parjadis, Quentin Cappart, Bistra Dilkina et al.

Lagrangian relaxation is a versatile mathematical technique employed to relax constraints in an optimization problem, enabling the generation of dual bounds to prove the optimality of feasible solutions and the design of efficient propagators in constraint programming (such as the weighted circuit constraint). However, the conventional process of deriving Lagrangian multipliers (e.g., using subgradient methods) is often computationally intensive, limiting its practicality for large-scale or time-sensitive problems. To address this challenge, we propose an innovative unsupervised learning approach that harnesses the capabilities of graph neural networks to exploit the problem structure, aiming to generate accurate Lagrangian multipliers efficiently. We apply this technique to the well-known Held-Karp Lagrangian relaxation for the travelling salesman problem. The core idea is to predict accurate Lagrangian multipliers and to employ them as a warm start for generating Held-Karp relaxation bounds. These bounds are subsequently utilized to enhance the filtering process carried out by branch-and-bound algorithms. In contrast to much of the existing literature, which primarily focuses on finding feasible solutions, our approach operates on the dual side, demonstrating that learning can also accelerate the proof of optimality. We conduct experiments across various distributions of the metric travelling salesman problem, considering instances with up to 200 cities. The results illustrate that our approach can improve the filtering level of the weighted circuit global constraint, reduce the optimality gap by a factor two for unsolved instances up to a timeout, and reduce the execution time for solved instances by 10%.

Adrien Rimélé, Philippe Grangier, Michel Gamache et al.

A Robotic Mobile Fulfillment System is a robotised parts-to-picker system that is particularly well-suited for e-commerce warehousing. One distinguishing feature of this type of warehouse is its high storage modularity. Numerous robots are moving shelves simultaneously, and the shelves can be returned to any open location after the picking operation is completed. This work focuses on the real-time storage allocation problem to minimise the travel time of the robots. An efficient -- but computationally costly -- Monte Carlo Tree Search method is used offline to generate high-quality experience. This experience can be learned by a neural network with a proper coordinates-based features representation. The obtained neural network is used as an action predictor in several new storage policies, either as-is or in rollout and supervised tree search strategies. Resulting performance levels depend on the computing time available at a decision step and are consistently better compared to real-time decision rules from the literature.

Léa Ricard, Guy Desaulniers, Andrea Lodi et al.

An important aspect of the quality of a public transport service is its reliability, which is defined as the invariability of the service attributes. Preventive measures taken during planning can reduce risks of unreliability throughout operations. In order to tackle reliability during the service planning phase, a key piece of information is the long-term prediction of the density of the travel time, which conveys the uncertainty of travel times. We introduce a reliable approach to one of the problems of service planning in public transport, namely the Multiple Depot Vehicle Scheduling Problem (MDVSP), which takes as input a set of trips and the probability density function (p.d.f.) of the travel time of each trip in order to output delay-tolerant vehicle schedules. This work empirically compares probabilistic models for the prediction of the conditional p.d.f. of the travel time, as a first step towards reliable MDVSP solutions. Two types of probabilistic models, namely similarity-based density estimation models and a smoothed Logistic Regression for probabilistic classification model, are compared on a dataset of more than 41,000 trips and 50 bus routes of the city of Montréal. The result of a vast majority of probabilistic models outperforms that of a Random Forests model, which is not inherently probabilistic, thus highlighting the added value of modeling the conditional p.d.f. of the travel time with probabilistic models. A similarity-based density estimation model using a $k$ Nearest Neighbors method and a Kernel Density Estimation predicted the best estimate of the true conditional p.d.f. on this dataset.

Adrien Rimélé, Philippe Grangier, Michel Gamache et al.

E-commerce with major online retailers is changing the way people consume. The goal of increasing delivery speed while remaining cost-effective poses significant new challenges for supply chains as they race to satisfy the growing and fast-changing demand. In this paper, we consider a warehouse with a Robotic Mobile Fulfillment System (RMFS), in which a fleet of robots stores and retrieves shelves of items and brings them to human pickers. To adapt to changing demand, uncertainty, and differentiated service (e.g., prime vs. regular), one can dynamically modify the storage allocation of a shelf. The objective is to define a dynamic storage policy to minimise the average cycle time used by the robots to fulfil requests. We propose formulating this system as a Partially Observable Markov Decision Process, and using a Deep Q-learning agent from Reinforcement Learning, to learn an efficient real-time storage policy that leverages repeated experiences and insightful forecasts using simulations. Additionally, we develop a rollout strategy to enhance our method by leveraging more information available at a given time step. Using simulations to compare our method to traditional storage rules used in the industry showed preliminary results up to 14\% better in terms of travelling times.

Minh Hoàng Hà, Tat Dat Nguyen, Thinh Nguyen Duy et al.

We consider a vehicle routing problem which seeks to minimize cost subject to time window and synchronization constraints. In this problem, the fleet of vehicles is categorized into regular and special vehicles. Some customers require both vehicles' services, whose starting service times at the customer are synchronized. Despite its important real-world application, this problem has rarely been studied in the literature. To solve the problem, we propose a Constraint Programming (CP) model and an Adaptive Large Neighborhood Search (ALNS) in which the design of insertion operators is based on solving linear programming (LP) models to check the insertion feasibility. A number of acceleration techniques is also proposed to significantly reduce the computational time. The computational experiments show that our new CP model finds better solutions than an existing CP-based ANLS, when used on small instances with 25 customers and with a much shorter running time. Our LP-based ALNS dominates the cp-ALNS, in terms of solution quality, when it provides solutions with better objective values, on average, for all instance classes. This demonstrates the advantage of using linear programming instead of constraint programming when dealing with a variant of vehicle routing problems with relatively tight constraints, which is often considered to be more favorable for CP-based methods.