arXiv subjects

Axel Parmentier

Publications and source records attributed to Axel Parmentier.

At least 37 records · Page 2Linked to original sources

arXiv2024

Learning-based Online Optimization for Autonomous Mobility-on-Demand Fleet Control

Autonomous mobility-on-demand systems are a viable alternative to mitigate many transportation-related externalities in cities, such as rising vehicle volumes in urban areas and transportation-related pollution. However, the success of these systems heavily depends on efficient and effective fleet control strategies. In this context, we study online control algorithms for autonomous mobility-on-demand systems and develop a novel hybrid combinatorial optimization enriched machine learning pipeline which learns online dispatching and rebalancing policies from optimal full-information solutions. We test our hybrid pipeline on large-scale real-world scenarios with different vehicle fleet sizes and various request densities. We show that our approach outperforms state-of-the-art greedy, and model-predictive control approaches with respect to various KPIs, e.g., by up to 17.1% and on average by 6.3% in terms of realized profit.

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arXiv2024

Combinatorial Optimization and Machine Learning for Dynamic Inventory Routing

We introduce a combinatorial optimization-enriched machine learning pipeline and a novel learning paradigm to solve inventory routing problems with stochastic demand and dynamic inventory updates. After each inventory update, our approach reduces replenishment and routing decisions to an optimal solution of a capacitated prize-collecting traveling salesman problem for which well-established algorithms exist. Discovering good prize parametrizations is non-trivial; therefore, we have developed a machine learning approach. We evaluate the performance of our pipeline in settings with steady-state and more complex demand patterns. Compared to previous works, the policy generated by our algorithm leads to significant cost savings, achieves lower inference time, and can even leverage contextual information.

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arXiv2023

Solving a Continent-Scale Inventory Routing Problem at Renault

This paper is the fruit of a partnership with Renault. Their backward logistic requires solving a continent-scale multi-attribute inventory routing problem (IRP). With an average of 30 commodities, 16 depots, and 600 customers spread across a continent, our instances are orders of magnitude larger than those in the literature. Existing algorithms do not scale. We propose a large neighborhood search (LNS). To make it work, (1) we generalize existing split delivery vehicle routing problem and IRP neighborhoods to this context, (2) we turn a state-of-the art matheuristic for medium-scale IRP into a large neighborhood, and (3) we introduce two novel perturbations: the reinsertion of a customer and that of a commodity into the IRP solution. We also derive a new lower bound based on a flow relaxation. In order to stimulate the research on large-scale IRP, we introduce a library of industrial instances. We benchmark our algorithms on these instances and make our code open-source. Extensive numerical experiments highlight the relevance of each component of our LNS.

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arXiv2023

Explainable Data-Driven Optimization: From Context to Decision and Back Again

Data-driven optimization uses contextual information and machine learning algorithms to find solutions to decision problems with uncertain parameters. While a vast body of work is dedicated to interpreting machine learning models in the classification setting, explaining decision pipelines involving learning algorithms remains unaddressed. This lack of interpretability can block the adoption of data-driven solutions as practitioners may not understand or trust the recommended decisions. We bridge this gap by introducing a counterfactual explanation methodology tailored to explain solutions to data-driven problems. We introduce two classes of explanations and develop methods to find nearest explanations of random forest and nearest-neighbor predictors. We demonstrate our approach by explaining key problems in operations management such as inventory management and routing.

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arXiv2023

Combinatorial Optimization enriched Machine Learning to solve the Dynamic Vehicle Routing Problem with Time Windows

With the rise of e-commerce and increasing customer requirements, logistics service providers face a new complexity in their daily planning, mainly due to efficiently handling same day deliveries. Existing multi-stage stochastic optimization approaches that allow to solve the underlying dynamic vehicle routing problem are either computationally too expensive for an application in online settings, or -- in the case of reinforcement learning -- struggle to perform well on high-dimensional combinatorial problems. To mitigate these drawbacks, we propose a novel machine learning pipeline that incorporates a combinatorial optimization layer. We apply this general pipeline to a dynamic vehicle routing problem with dispatching waves, which was recently promoted in the EURO Meets NeurIPS Vehicle Routing Competition at NeurIPS 2022. Our methodology ranked first in this competition, outperforming all other approaches in solving the proposed dynamic vehicle routing problem. With this work, we provide a comprehensive numerical study that further highlights the efficacy and benefits of the proposed pipeline beyond the results achieved in the competition, e.g., by showcasing the robustness of the encoded policy against unseen instances and scenarios.

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arXiv2023

Learning structured approximations of combinatorial optimization problems

Machine learning pipelines that include a combinatorial optimization layer can give surprisingly efficient heuristics for difficult combinatorial optimization problems. Three questions remain open: which architecture should be used, how should the parameters of the machine learning model be learned, and what performance guarantees can we expect from the resulting algorithms? Following the intuitions of geometric deep learning, we explain why equivariant layers should be used when designing such pipelines, and illustrate how to build such layers on routing, scheduling, and network design applications. We introduce a learning approach that enables to learn such pipelines when the training set contains only instances of the difficult optimization problem and not their optimal solutions, and show its numerical performance on our three applications. Finally, using tools from statistical learning theory, we prove a theorem showing the convergence speed of the estimator. As a corollary, we obtain that, if an approximation algorithm can be encoded by the pipeline for some parametrization, then the learned pipeline will retain the approximation ratio guarantee. On our network design problem, our machine learning pipeline has the approximation ratio guarantee of the best approximation algorithm known and the numerical efficiency of the best heuristic.

cs.RO↗

arXiv2023

Electric Vehicle Fleets: Scalable Route and Recharge Scheduling through Column Generation

The rise of battery-powered vehicles has led to many new technical and methodological hurdles. Among these, the efficient planning of an electric fleet to fulfill passenger transportation requests still represents a major challenge. This is due to the specific constraints of electric vehicles, bound by their battery autonomy and necessity of recharge planning, and the large scale of the operations, which challenges existing optimization algorithms. The purpose of this paper is to introduce a scalable column generation approach for routing and scheduling in this context. Our algorithm relies on four main ingredients: (i) a multigraph reformulation of the problem based on a characterization of non-dominated charging arcs, (ii) an efficient bi-directional pricing algorithm using tight backward bounds, (iii) sparsification approaches permitting to decrease the size of the subjacent graphs dramatically, and (iv) a diving heuristic, which locates near-optimal solutions in a fraction of the time needed for a complete branch-and-price. Through extensive computational experiments, we demonstrate that our approach significantly outperforms previous algorithms for this setting, leading to accurate solutions for problems counting several hundreds of requests.

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arXiv2022

Learning with Combinatorial Optimization Layers: a Probabilistic Approach

Combinatorial optimization (CO) layers in machine learning (ML) pipelines are a powerful tool to tackle data-driven decision tasks, but they come with two main challenges. First, the solution of a CO problem often behaves as a piecewise constant function of its objective parameters. Given that ML pipelines are typically trained using stochastic gradient descent, the absence of slope information is very detrimental. Second, standard ML losses do not work well in combinatorial settings. A growing body of research addresses these challenges through diverse methods. Unfortunately, the lack of well-maintained implementations slows down the adoption of CO layers. In this paper, building upon previous works, we introduce a probabilistic perspective on CO layers, which lends itself naturally to approximate differentiation and the construction of structured losses. We recover many approaches from the literature as special cases, and we also derive new ones. Based on this unifying perspective, we present InferOpt.jl, an open-source Julia package that 1) allows turning any CO oracle with a linear objective into a differentiable layer, and 2) defines adequate losses to train pipelines containing such layers. Our library works with arbitrary optimization algorithms, and it is fully compatible with Julia's ML ecosystem. We demonstrate its abilities using a pathfinding problem on video game maps as guiding example, as well as three other applications from operations research.

stat.ML↗

arXiv2022

Linear lexicographic optimization and preferential bidding system

Some airlines use the preferential bidding system to construct the schedules of their pilots. In this system, the pilots bid on the different activities and the schedules that lexicographically maximize the scores of the pilots according to their seniority are selected. A sequential approach to solve this maximization problem is natural: the problem is first solved with the bids of the most senior pilot; then it is solved with those of the second most senior without decreasing the score of the most senior, and so on. The literature admits that the structure of the problem somehow imposes such an approach. The problem can be modeled as an integer linear lexicographic program. We propose a new exact method, which relies on column generation for solving its continuous relaxation. To design this column generation, we prove that bounded linear lexicographic programs admit "primal-dual" feasible bases and we show how to compute such bases efficiently. Another contribution on which our exact method relies consists in the extension of standard tools for resource-constrained longest path problems to their lexicographic versions. This is useful in our context since the generation of new columns is modeled as a lexicographic resource-constrained longest path problem. Numerical experiments show that this new method is already able to solve industrial instances provided by Air France, with up to 150 pilots. By adding a last ingredient in the resolution of the longest path problems, which exploits the specificity of the preferential bidding system, the method achieves for these instances computational times that are compatible with operational constraints.

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arXiv2022

Future memories are not needed for large classes of POMDPs

Optimal policies for partially observed Markov decision processes (POMDPs) are history-dependent: Decisions are made based on the entire history of observation. Memoryless policies, which take decisions based on the last observation only, are generally considered useless in the literature because we can construct POMDP instances for which optimal memoryless policies are arbitrarily worse than history-dependent ones. Our purpose is to challenge this belief. We show that optimal memoryless policies can be computed efficiently using mixed integer linear programming (MILP), and perform reasonably well on a wide range of instances from the literature. When strengthened with valid inequalities, the linear relaxation of this MILP provides high quality upper-bounds on the value of an optimal history dependent policy. Furthermore, when used with a finite horizon POMDP problem with memoryless policies as rolling optimization problem, a model predictive control approach leads to an efficient history-dependent policy, which we call the short memory in the future (SMF) policy. Basically, the SMF policy leverages these memoryless policies to build an approximation of the Bellman value function. Numerical experiments show the efficiency of our approach on benchmark instances from the literature.

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arXiv2021

Optimal Counterfactual Explanations in Tree Ensembles

Counterfactual explanations are usually generated through heuristics that are sensitive to the search's initial conditions. The absence of guarantees of performance and robustness hinders trustworthiness. In this paper, we take a disciplined approach towards counterfactual explanations for tree ensembles. We advocate for a model-based search aiming at "optimal" explanations and propose efficient mixed-integer programming approaches. We show that isolation forests can be modeled within our framework to focus the search on plausible explanations with a low outlier score. We provide comprehensive coverage of additional constraints that model important objectives, heterogeneous data types, structural constraints on the feature space, along with resource and actionability restrictions. Our experimental analyses demonstrate that the proposed search approach requires a computational effort that is orders of magnitude smaller than previous mathematical programming algorithms. It scales up to large data sets and tree ensembles, where it provides, within seconds, systematic explanations grounded on well-defined models solved to optimality.

cs.LG↗

arXiv2021

Learning to solve the single machine scheduling problem with release times and sum of completion times

In this paper, we focus on the solution of a hard single machine scheduling problem by new heuristic algorithms embedding techniques from machine learning field and scheduling theory. These heuristics transform an instance of the hard problem into an instance of a simpler one solved to optimality. The obtained schedule is then transposed to the original problem. Computational experiments show that they are competitive with state-of-the-art heuristics, notably on large instances.

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arXiv2020

Integer programming for weakly coupled stochastic dynamic programs with partial information

This paper introduces algorithms for problems where a decision maker has to control a system composed of several components and has access to only partial information on the state of each component. Such problems are difficult because of the partial observations, and because of the curse of dimensionality that appears when the number of components increases. Partially observable Markov decision processes (POMDPs) have been introduced to deal with the first challenge, while weakly coupled stochastic dynamic programs address the second. Drawing from these two branches of the literature, we introduce the notion of weakly coupled POMDPs. The objective is to find a policy maximizing the total expected reward over a finite horizon. Our algorithms rely on two ingredients. The first, which can be used independently, is a mixed integer linear formulation for generic POMDPs that computes an optimal memoryless policy. The formulation is strengthened with valid cuts based on a probabilistic interpretation of the dependence between random variables, and its linear relaxation provide a practically tight upper bound on the value of an optimal history-dependent policies. The second is a collection of mathematical programming formulations and algorithms which provide tractable policies and upper bounds for weakly coupled POMDPs. Lagrangian relaxations, fluid approximations, and almost sure constraints relaxations enable to break the curse of dimensionality. We test our generic POMDPs formulations on benchmark instance forms the literature, and our weakly coupled POMDP algorithms on a maintenance problem. Numerical experiments show the efficiency of our approach.

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arXiv2019

Aircraft routing and crew pairing: updated algorithms at Air France

Aircraft routing and crew pairing problems aims at building the sequences of flight legs operated respectively by airplanes and by crews of an airline. Given their impact on airlines operating costs, both have been extensively studied for decades. Our goal is to provide reliable and easy to maintain frameworks for both problems at Air France. We propose simple approaches to deal with Air France current setting. For routing, we introduce a compact MIP formulation that can be solved by current MIP solvers in at most a few minutes even on Air France largest instances. Regarding crew pairing, we provide a standard methodology to model the column generation pricing subproblem within a new resource constrained shortest path framework recently introduced by the first author. This new framework, which can be used as a black-box, leverages on bounds on paths' resource to discard partial solutions and speed-up the resolution. The resulting approach enables to solve to optimality Air France largest instances. Recent literature has focused on integrating aircraft routing and crew pairing problems. As a side result, we have been able to solve to near optimality large industrial instances of the integrated problem by combining the aforementioned algorithms within a simple cut generating method.

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arXiv2019

Integer programming on the junction tree polytope for influence diagrams

Influence Diagrams (ID) are a flexible tool to represent discrete stochastic optimization problems, including Markov Decision Process (MDP) and Partially Observable MDP as standard examples. More precisely, given random variables considered as vertices of an acyclic digraph, a probabilistic graphical model defines a joint distribution via the conditional distributions of vertices given their parents. In ID, the random variables are represented by a probabilistic graphical model whose vertices are partitioned into three types : chance, decision and utility vertices. The user chooses the distribution of the decision vertices conditionally to their parents in order to maximize the expected utility. Leveraging the notion of rooted junction tree, we present a mixed integer linear formulation for solving an ID, as well as valid inequalities, which lead to a computationally efficient algorithm. We also show that the linear relaxation yields an optimal integer solution for instances that can be solved by the "single policy update", the default algorithm for addressing IDs.

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arXiv2019

Linear Programming for Decision Processes with Partial Information

Markov Decision Processes (MDPs) are stochastic optimization problems that model situations where a decision maker controls a system based on its state. Partially observed Markov decision processes (POMDPs) are generalizations of MDPs where the decision maker has only partial information on the state of the system. Decomposable POMDPs are specific cases of POMDPs that enable one to model systems with several components. Such problems naturally model a wide range of applications such as predictive maintenance. Finding an optimal policy for a POMDP is PSPACE-hard and practically challenging. We introduce a mixed integer linear programming (MILP) formulation for POMDPs restricted to the policies that only depend on the current observation, as well as valid inequalities that are based on a probabilistic interpretation of the dependence between variables. The linear relaxation provides a good bound for the usual POMDPs where the policies depend on the full history of observations and actions. Solving decomposable POMDPs is especially challenging due to the curse of dimensionality. Leveraging our MILP formulation for POMDPs, we introduce a linear program based on ``fluid formulation'' for decomposable POMDPs, that provides both a bound on the optimal value and a practically efficient heuristic to find a good policy. Numerical experiments show the efficiency of our approaches to POMDPs and decomposable POMDPs.

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arXiv2019

Two generalizations of Markov blankets

In a probabilistic graphical model on a set of variables $V$, the Markov blanket of a random vector $B$ is the minimal set of variables conditioned to which $B$ is independent from the remaining of the variables $V \backslash B$. We generalize Markov blankets to study how a set $C$ of variables of interest depends on~$B$. Doing that, we must choose if we authorize vertices of $C$ or vertices of $V \backslash C$ in the blanket. We therefore introduce two generalizations. The Markov blanket of $B$ in $C$ is the minimal subset of $C$ conditionally to which $B$ and $C$ are independent. It is naturally interpreted as the inner boundary through which $C$ depends on $B$, and finds applications in feature selection. The Markov blanket of $B$ in the direction of $C$ is the nearest set to $B$ among the minimal sets conditionally to which ones $B$ and $C$ are independent, and finds applications in causality. It is the outer boundary of $B$ in the direction of $C$. We provide algorithms to compute them that are not slower than the usual algorithms for finding a d-separator in a directed graphical model. All our definitions and algorithms are provided for directed and undirected graphical models.

math.PR↗

arXiv2018

Ground staff shift planning under delay uncertainty at Air France

Ground staff agents of airlines operate many jobs at airports such as passengers check-in, planes cleaning, etc. Shift planning aims at building the sequences of jobs operated by ground staff agents, and have been widely studied given its impact on operating costs. As these jobs are closely related to flights arrivals and departures, flights delays disrupt ground staff schedules, which leads to high additional costs. Our goal is to provide a solution for shift planning at Air France that takes into account these additional costs. We therefore introduce a stochastic version of the shift planning problem that takes into accounts the cost of disruptions due to delay, and a column generation approach to solve it. The key element of our column generation is the algorithm for the pricing subproblem, which we model as a stochastic resource constrained shortest path problem. Numerical results on Air France industrial instances prove the relevance of the shift planning problem and the efficiency of the solution method. The column generation can solve to optimality instances with up to two hundred and fifty jobs. Moving from the deterministic problem to the stochastic one including delay costs enables to reduce the total operating costs by 3% to 5% on our instances.

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