arXiv subjects

Nicolas Dupin

Publications and source records attributed to Nicolas Dupin.

12 recordsLinked to original sources

arXiv2025

The Bi-objective Electric Autonomous Dial-a-Ride Problem

The electric autonomous dial-a-ride problem (E-ADARP) introduces electric, autonomously driving vehicles and their unique requirements into the classic dial-a-ride problem, where people are transported between pickup and drop-off locations. Next to an electric autonomous vehicle fleet, in the literature, a weighted-sum objective function, which combines the classic routing cost-oriented objective with a user-oriented objective function, has usually been considered. The user-oriented objective function minimizes the total excess user ride time. In this work, we treat them as two separate objective functions, which are optimized concurrently. In order to address the resulting bi-objective E-ADARP, we develop a novel exact framework (called fragment-based checker), whose core part is a smart ``select-and-check" algorithm that iteratively constructs feasible solutions using fragments. Several enhancements are proposed to enforce the computational efficiency of the proposed method. In the computational experiments, we evaluate several variants of our checker algorithm by leveraging a previously developed branch-and-price algorithm. We benchmark the checker-based framework against state-of-the-art criterion space frameworks as well as a generalized branch-and-price algorithm. Numerical results on both bi-objective DARP and E-ADARP instances demonstrate the effectiveness of the proposed framework. With our proposed approaches, 21 out of 38 instances are solved optimally, where small-to-medium-sized instances are solved within seconds. On larger-scale instances, especially those requiring high battery end levels are computationally challenging to solve, our approaches provide high-quality approximations of the Pareto frontiers. Efficient solutions with varying energy restrictions are compared and we obtain valuable managerial insights for different kinds of service providers.

math.OC↗

arXiv2024

A Branch-and-Price Algorithm for the Electric Autonomous Dial-A-Ride Problem

The Electric Autonomous Dial-A-Ride Problem (E-ADARP) consists in scheduling a fleet of electric autonomous vehicles to provide ride-sharing services for customers that specify their origins and destinations. The E-ADARP differs from the classical DARP in two aspects: (i) a weighted-sum objective that minimizes both total travel time and total excess user ride time; (ii) the employment of electric autonomous vehicles and a partial recharging policy. This paper presents a highly-efficient labeling algorithm, which is integrated into Branch-and-Price (B&P) algorithms to solve the E-ADARP. To handle (i), we introduce a fragment-based representation of paths. A novel approach is invoked to abstract fragments to arcs while ensuring excess-user-ride-time optimality. We then construct a new graph that preserves all feasible routes of the original graph by enumerating all feasible fragments, abstracting them to arcs, and connecting them with each other, depots, and recharging stations in a feasible way. On the new graph, partial recharging (ii) is tackled exactly by tailored Resource Extension Functions (REFs). We apply strong dominance rules and constant-time feasibility checks to compute the shortest paths efficiently. These methods construct the first labeling algorithm that can deal with minimizing (excess) user ride time. In the computational experiments, the B&P algorithm achieves optimality in 71 out of 84 instances. Remarkably, among these instances, 50 were solved optimally at the root node without branching. We identify 26 new best solutions, improve 30 previously reported lower bounds, and provide 17 new lower bounds for large-scale instances with up to 8 vehicles and 96 requests. In total 42 new best solutions are generated on previously solved and unsolved instances.

math.OC↗

arXiv2023

Polynomial algorithms for p-dispersion problems in a planar Pareto Front

In this paper, p-dispersion problems are studied to select $p\geqslant 2$ representative points from a large 2D Pareto Front (PF), solution of bi-objective optimization. Four standard p-dispersion variants are considered. A novel variant, Max-Sum-Neighbor p-dispersion, is introduced for the specific case of a 2D PF. Firstly, $2$-dispersion and $3$-dispersion problems are proven solvable in $O(n)$ time in a 2D PF. Secondly, dynamic programming algorithms are designed for three p-dispersion variants, proving polynomial complexities in a 2D PF. Max-min p-dispersion is solvable in $O(pn\log n)$ time and $O(n)$ memory space. Max-Sum-Neighbor p-dispersion is proven solvable in $O(pn^2)$ time and{$O(n)$} space. Max-Sum-min p-dispersion is solvable in $O(pn^3)$ time and $O(pn^2)$ space, this complexity holds also in 1D, proving for the first time that Max-Sum-min p-dispersion is polynomial in 1D. Furthermore, properties of these algorithms are discussed for an efficient implementation {and for a practical application inside bi-objective meta-heuristics.

cs.DS↗

arXiv2023

Partial k-means to avoid outliers, mathematical programming formulations, complexity results

A well-known bottleneck of Min-Sum-of-Square Clustering (MSSC, the celebrated $k$-means problem) is to tackle the presence of outliers. In this paper, we propose a Partial clustering variant termed PMSSC which considers a fixed number of outliers to remove. We solve PMSSC by Integer Programming formulations and complexity results extending the ones from MSSC are studied. PMSSC is NP-hard in Euclidean space when the dimension or the number of clusters is greater than $2$. Finally, one-dimensional cases are studied: Unweighted PMSSC is polynomial in that case and solved with a dynamic programming algorithm, extending the optimality property of MSSC with interval clustering. This result holds also for unweighted $k$-medoids with outliers. A weaker optimality property holds for weighted PMSSC, but NP-hardness or not remains an open question in dimension one.

cs.CC↗

arXiv2023

A Deterministic Annealing Local Search for the Electric Autonomous Dial-A-Ride Problem

This paper investigates the Electric Autonomous Dial-A-Ride Problem (E-ADARP), which consists in designing a set of minimum-cost routes that accommodates all customer requests for a fleet of Electric Autonomous Vehicles (EAVs). Problem-specific features of the E-ADARP include: (i) the employment of EAVs and a partial recharging policy; (ii) the weighted-sum objective function that minimizes the total travel time and the total excess user ride time. In this work, we propose a Deterministic Annealing (DA) algorithm and provide the first heuristic results for the static E-ADARP. Partial recharging (i) is handled by an exact route evaluation scheme of linear time complexity. To tackle (ii), we propose a new method that allows effective computations of minimum excess user ride time by introducing a fragment-based representation of paths. These two methods compose an exact and efficient optimization of excess user ride time for a generated E-ADARP route. To validate the performance of the DA algorithm, we compare our algorithm results to the best-reported Branch-and-Cut (B\&C) algorithm results on existing instances. Our algorithm provides 25 new best solutions and 45 equal solutions on 84 existing instances. To test the algorithm performance on larger-sized instances, we establish new instances with up to 8 vehicles and 96 requests, and we provide 19 new solutions for these instances. Our final investigation extends the state-of-the-art model and explores the effect of allowing multiple visits to recharging stations. This relaxation can efficiently improve the solution's feasibility and quality.

math.OC↗

arXiv2021

An exact dynamic programming algorithm, lower and upper bounds, applied to the large block sale problem

In this article, we address a class of non convex, integer, non linear mathematical programs using dynamic programming. The mathematical program considered, whose properties are studied in this article, may be used to model the optimal liquidation problem of a single asset portfolio, held in a very large quantity, in a low volatility and perfect memory market, with few market participants. In this context, the Portfolio Manager's selling actions convey information to market participants, which in turn lower bid prices and further penalize the liquidation proceeds we attempt to maximize. We show the problem can be solved exactly using Dynamic Programming (DP) in polynomial time. However, exact resolution is only efficient for small instances. For medium size and large instances, we introduce dedicated heuristics which provide thin admissible solutions, hence tight lower bounds for the initial problem. We also benchmark them against a commercial solver, such as LocalSolver [7]. We are also interested in the continuously relaxed problem, which is non convex. Firstly, we use continuous solutions, obtained by free solver NLopt [26] and transform them into thin admissible solutions of the discrete problem. Secondly, we provide, under some convexity assumptions, an upper bound for the continuous relaxation, and hence for the initial (integer) problem. Numerical experiments confirm the quality of proposed heuristics (lower bounds), which often reach the optimal, or prove very tight, for small and medium size instances, with a very fast CPU time. Our upper bound, however, is not tight.

cs.DM↗

arXiv2020

Matheuristics to optimize refueling and maintenance planning of nuclear power plants

Planning the maintenance of nuclear power plants is a complex optimization problem, involving a joint optimization of maintenance dates, fuel constraints and power production decisions. This paper investigates Mixed Integer Linear Programming (MILP) matheuristics for this problem, to tackle large size instances used in operations with a time scope of five years, and few restrictions with time window constraints for the latest maintenance operations. Several constructive matheuristics and a Variable Neighborhood Descent local search are designed. The matheuristics are shown to be accurately effective for medium and large size instances. The matheuristics give also results on the design of MILP formulations and neighborhoods for the problem. Contributions for the operational applications are also discussed. It is shown that the restriction of time windows, which was used to ease computations, induces large over-costs and that this restriction is not required anymore with the capabilities of matheuristics or local search to solve such size of instances. Our matheuristics can be extended to a bi-objective optimization extension with stability costs, for the monthly re-optimization of the maintenance planning in the real-life application.

cs.AI↗

arXiv2020

k-medoids and p-median clustering are solvable in polynomial time for a 2d Pareto front

This paper examines a common extension of k-medoids and k-median clustering in the case of a two-dimensional Pareto front, as generated by bi-objective optimization approaches. A characterization of optimal clusters is provided, which allows to solve the optimization problems to optimality in polynomial time using a common dynamic programming algorithm. More precisely, having $N$ points to cluster in $K$ subsets, the complexity of the algorithm is proven in $O(N^3)$ time and $O(K.N)$ memory space when $K\geqslant 3$, cases $K=2$ having a time complexity in $O(N^2)$. Furthermore, speeding-up the dynamic programming algorithm is possible avoiding useless computations, for a practical speed-up without improving the complexity. Parallelization issues are also discussed, to speed-up the algorithm in practice.

cs.CC↗

arXiv2020

Long term planning of military aircraft flight and maintenance operations

We present the Flight and Maintenance Planning (FMP) problem in its military variant and applied to long term planning. The problem has been previously studied for short- and medium-term horizons only. We compare its similarities and differences with previous work and prove its complexity. We generate scenarios inspired by the French Air Force fleet. We formulate an exact Mixed Integer Programming (MIP) model to solve the problem in these scenarios and we analyse the performance of the solving method under these circumstances. A heuristic was built to generate fast feasible solutions, that in some cases were shown to help warm-start the model.

cs.AI↗

arXiv2019

Column generation for the discrete Unit Commitment problem with min-stop ramping constraints

The discrete unit commitment problem with min-stop ramping constraints optimizes the daily production of thermal power plants (coal, gas, fuel units). For this problem, compact Integer Linear Programming (ILP) formulations have been designed to solve exactly small instances and heuristically real-size instances. This paper investigates whether Dantzig-Wolfe reformulation allows to improve the previous exact method and matheuristics. The extended ILP formulation is presented with the column generation algorithm to solve its linear relaxation. The experimental results show that the Dantzig-Wolfe reformulation does not improve the quality of the linear relaxation of the tightest compact ILP formulations. Computational experiments suggest also a conjecture which would explain such result: the compact ILP formulation of min-stop ramping constraints would be tight. Such results validate the quality of the exact methods and matheuristics based on compact ILP formulations previously designed.

math.OC↗

arXiv2019

Planar p-center problems are solvable in polynomial time when clustering a Pareto Front

This paper is motivated by real-life applications of bi-objective optimization. Having many non dominated solutions, one wishes to cluster the Pareto front using Euclidian distances. The p-center problems, both in the discrete and continuous versions, are proven solvable in polynomial time with a common dynamic programming algorithm. Having $N$ points to partition in $K\geqslant 3$ clusters, the complexity is proven in $O(KN\log N)$ (resp $O(KN\log^2 N)$) time and $O(KN)$ memory space for the continuous (resp discrete) $K$-center problem. $2$-center problems have complexities in $O(N\log N)$. To speed-up the algorithm, parallelization issues are discussed. A posteriori, these results allow an application inside multi-objective heuristics to archive partial Pareto Fronts.

cs.CG↗

arXiv2018

Dual heuristics and new dual bounds to schedule the maintenances of nuclear power plants

The EURO/ROADEF 2010 Challenge aimed to schedule the maintenance and refueling operations of French nuclear power plants, ranking the approaches in competition for the quality of primal solutions. This paper justifies the high quality of the best solutions computing dual bounds with dual heuristics. A first step designs several Mixed Integer Programming (MIP) relaxations with different compromises between computation time and quality of dual bounds. To deal with smaller MIPs, we prove how reductions in the number of time steps and scenarios can guarantee dual bounds for the whole problem of the Challenge. Several sets of dual bounds are computable, improving significantly the former best dual bounds of the literature. Intermediate results allow also a better understanding of the problem and offer perspectives to improve some approaches of the Challenge.

math.OC↗