Mathieu LACROIX Laboratoire LAMSADE Université Paris Dauphine Bureau C605 Place du Maréchal de Lattre de Tassigny 75775 Paris cedex 16 - France Tél : +33 (0)1 44 05 48 53 E-mail : lacroix@lamsade.dauphine.fr url : http://epoc.isima.fr/~lacroix/ Date of birth : April 28, 1982 Nationality : French Degrees held 2005 - Ph.D. Student in Combinatorial Optimization Laboratoire d Informatique, de Modélisation et d Optimisation des Systèmes (LIMOS) Subject of the thesis : The preemptive pickup and delivery problem : complexity, models and polyhedra. Supervisors : A. R. Mahjoub and A. Quilliot Co-supervisor : H. Kerivin 2004-2005 Master degree in Operational Research, rank : 1st Laboratoire d Informatique, de Modélisation et d Optimisation des Systèmes (LIMOS) Subject : Polyhedral study of the preemptive pickup and delivery with split loads Supervisors : A. R. Mahjoub and A. Quilliot Co-supervisor : H. Kerivin 2000-2004 Bs. in Computer Science and Operational Research, rank : 1st 2000 Baccalauréat Mathematics and physics, Lycée Blaise Pascal, Clermont-Ferrand, France Professional experiences 2008-2009 Lecture, Department of Mathematics and Computer Science, Université Paris- Dauphine, France. 2005-2008 Researcher, Laboratoire LIMOS, Université Blaise-Pascal, Clermont II, Clermont- Ferrand, France. 1
Research Activities My research is related to the field of Combinatorial Optimization. It concerns the so-called pickup and delivery problem. We have first studied the complexity of some related problems and given integer programming formulations for some variants of the problem. Now, we are studying the convex hull of the solutions of the problem. Our aim is to develop efficient cutting plane based methods for solving big size instances of the problem. Combinatorial Optimization is an area of computer science and applied mathematics. It uses combinatorial, linear programming and algorithmic technics to solve optimization problems with discrete structures (usually a graph). These problems consist in finding the best element in a finite set. Several problems can be formulated as combinatorial optimization problems, especially transportation ones. However, even if the number of solutions is finite, the combinatorial explosion prevents from solving these problems by enumeration. Thus, it is necessary to use other technics which are more efficient. One of the most efficient methods to solve combinatorial optimization problems is the so-called polyhedral method. It consists in describing the convex hull of the solutions of the problem with the use of linear inequalities, which leads to a linear program that can be solved in polynomial time. Unfortunatelly, if the problem is NP-complete, there is a little chance to describe the convex hull. However, a partial description allows to design efficient branch and cut algorithms to solve the problem. Pickup and delivery problems. Transportation problems form a wilde area of operations research problems. This can be explained by the more and more important part of transportation in economy, due to globalization and intensification of exchanges of products and persons. Therefore, it is very important to improve algorithmic technics solving transportation problems and to consider more and more variants of such problems to match as close as possible real-world applications. To this aim, my PhD subject is concerned with preemptive pickup and delivery problems in which demands can be temporarily dropped anywhere in the network to be reloaded afterwards. Branch and cut algorithms. We have first considered the preemptive pickup and delivery problem with split demands, that is, when these latter can also be carried through different paths. We formulated this problem using two mixed-integer linear problems based on a space-time graph. We solved these two formulations with a branch-and-cut algorithm. This work led to the publications [1],[2]. Complexity and Modeling. We have particularly focused on the case when only one vehicle is available and the demands cannot be splitted (but they can still be temporarily dropped anywhere). This problem, called the Single-vehicle Preemptive Pickup and Delivery Problem, is still NP-hard. We have first studied the structure of the solutions of the problem. We have shown that a solution has to be described by the set of arcs traversed by the vehicle, the order in which these arcs are traversed and the set of arcs of the demand paths. We have also proved that these informations are necessary to check in polynomial time if a solution is feasible. This means that if one of these informations is missing, then checking if a solution is feasible is an NP-complete problem. Finally, we have given an integer linear programming formulation for this problem. We have also studied this problem when only one demand is carried at the same time (unitary case). The problem remains NP-hard in this case. However, we have shown that the information concerning the sequence of arcs traversed by the vehicle is no more necessary to check in polynomial time if a solution is feasible. Indeed, in this case, the order on the arcs can be obtained from the sets of arcs traversed by the vehicle and the demands. We have also introduced a new formulation for this 2
variant of the single-vehicle preemptive pickup and delivery problem based on these results. This work on complexity and modeling gave rise to the submitted articles [3], [4]. Polyhedral study and algorithms. Currently, we are studying the facial structure of the polytope of the unitary single-vehicle preemptive pickup and delivery problem. We have given necessary and sufficient conditions for the inequalities of the formulation to define facets. Based on these results, we have developed a branch-and-cut algorithm to solve this problem [5]. In order to improve this algorithm, we have also devised a metaheuristic [6] for computing an upper bound (allowing to prun some nodes in the branch tree). The originality of this metaheuristic comes from the fact that the solutions are coded using a bi-tree. Indeed, a deep study of the properties and structures of the optimal solutions has allowed us to represent every solution thanks to a bi-tree. Structural analysis of algebraic-differential systems with conditional equations Currently, I am collaborating with A. R. Mahjoub and S. Martin within an ANR project. The aim of the project is to develop a parallel algorithm for solving algebraic-differential systems when the system has conditional equations. One of the steps in the resolution of the system is the analysis of the structure of the system. This consists in checking if the system is feasible in all the configurations of the conditions. We have shown that this problem reduces to the Perfect Matching-Free Induced Subgraph Problem in Bipartite Graphs that can be stated as follows. Consider a bipartite graph given by the vertex sets V 1 and V 2 where the set of edges is the union of two sets, called True and False respectively (edges may belong to the two sets). Given a true or false value for each vertex of V 1, we consider the graph induced by the set of edges belonging to True incident to a vertex of V 1 with a true value and the set of edges belonging to False incident to a vertex of V 1 with a false value. The problem consists then in checking if there exists an assignment of true and false values to the vertices of V 1 so that the induced subgraph does not contain a perfect matching. We have first shown that this problem is NP-complete. We have also given formulations for the problem in terms of integer linear programs. In addition, we have shown that this problem reduces to the stable set problem in an auxiliary graph [7]. We are now studying the polyhedra of these formulations for devising a branch-and-cut algorithm for the problem. Future research In what concerns the unitary single-vehicle preemptive pickup and delivery problem, we want to more investigate the polyhedral structure of the problem. Indeed, we want to identify further facets for this problem in order to enforce the linear relaxation and obtain strenghtened lower bounds for the branch-and-cut algorithm. We also want to give a complete linear description of the associated polytope when the graph is a cactus or a serie-parallel graph. We already have some preliminary results for the case when the graph is a circuit. We also want to study an arc-path formulation for the problem and thus develop a branch-and-cut-and-price algorithm using this approach. It would also be interesting to consider other variants of the preemptive pickup and delivery problem. A first extension we would like to investigate is when several vehicles are used. We would also like to continue our polyhedral study of the formulation related to the problem of the structural analysis of the algebraic-differential systems. 3
Publications International journals [1] H. Kerivin, M. Lacroix, A. R. Mahjoub and A. Quilliot, The splittable pickup and delivery problem with reloads, European Journal of Industrial Engineering (EJIE), Vol. 2, N. 2, pages 112-133 (2008). Proceedings [2] H. Kerivin, M. Lacroix, A. R. Mahjoub and A. Quilliot, The capacitated vehicle routing problem with reloads, Proceedings of International Conference on Service System and Service Management (IEEE), pages 1513-1518 (2006). Submitted papers [3] H. Kerivin, M. Lacroix and A. R. Mahjoub, On the complexity of the Eulerian closed walk with precedence path constraints problem, submitted to ACM Transactions on Algorithms. [4] H. Kerivin, M. Lacroix and A. R. Mahjoub, Models for the single-vehicle preemptive pickup and delivery problem, submitted to Journal of Combinatorial Optimization (JOCO). Papers in preparation [5] H. Kerivin, M. Lacroix and A. R. Mahjoub, Polyhedral results for the unitary preemptive pickup and delivery problem, in preparation. [6] H. Kerivin, M. Lacroix, A. Quilliot and H. Toussaint, A heuristic for solving the unitary preemptive pickup and delivery problem, in preparation. [7] M. Lacroix, A. R. Mahjoub and S. Martin, On the structural analysis problem of algebraicdifferential systems, in preparation. Communications at conferences [8] H. Kerivin, M. Lacroix and A. R. Mahjoub, Le problème de la grue préemptif asymétrique : polyèdre et algorithme de coupes et branchements, 10 ème Congrès de la Société Française de Recherche Opérationnelle et d Aide à la Décision (ROADEF2009), february 2009, Nancy, France. [9] H. Kerivin, M. Lacroix and A. R. Mahjoub, The single-vehicle preemptive pickup and delivery problem, 13 th Combinatorial Optimization Workshop (on invitation), january 2009, Aussois, France. [10] H. Kerivin, M. Lacroix and A. R. Mahjoub, Le problème de cueillettes et livraisons préemptif avec un véhicule, 9 ème Congrès de la Société Française de Recherche Opérationnelle et d Aide à la Décision (ROADEF2008), february 2008, Clermont-Ferrand, France. [11] H. Kerivin, M. Lacroix and A. R. Mahjoub, The single-vehicle preemptive pickup and delivery problem, special session Combinatorial Optimization, International Conference on nonconvex 4
programming : LOCAL and GLOBAL approaches (NCP07), december 2007, Rouen, France. [12] H. Kerivin, M. Lacroix and A. R. Mahjoub, Formulations pour le problème de cueillettes et livraisons préemptif avec un véhicule, 4 ème Journées Polyèdre et Optimisation Combinatoire (JPOC4), june 2007, Évry, France. [13] H. Kerivin, M. Lacroix, A. R. Mahjoub and A. Quilliot, The capacitated vehicle routing problem with reloads, International Conference Service System and Service Management (IEEE), october 2006, Troyes, France. [14] H. Kerivin, M. Lacroix, A. R. Mahjoub and A. Quilliot, Le problème de trajets de véhicules avec déchargements/rechargements sous contraintes de capacités, 6 ème Congrès de la Société Française de Recherche Opérationnelle et d Aide à la Décision (ROADEF2005), february 2005, Lille, France. Other activities Co-supervisor of a master degree thesis on the single-vehicle preemptive pickup and delivery problem and of a training course about graph libraries (Boost Graph Library and Coin Graph Classes). Computer science languages and Optimization softwares Operating systems : Linux, Unix, Windows Programming languages : C, C++, JAVA, Pascal, SQL, Html/CSS Operational research : COIN-OR, Cplex, Boost Graph Library (BGL) Foreign languages English : good References Hervé Kerivin, Assistant Professor, Department of Mathematical Sciences, Clemson University, O-326 Martin Hall, Clemson, SC 29634, USA Tél : 864-656-0662 E-mail : kerivin@clemson.edu A. Ridha Mahjoub, Professor, Laboratoire LAMSADE, Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75775 Paris cedex 16 Tél : 01 44 05 48 96 E-mail : mahjoub@lamsade.dauphine.fr Alain Quilliot, Professor and director of Laboratoire LIMOS, Université Blaise Pascal, Clermont II, Complexe scientifique des Cézeaux, 63 173 Aubière, Cedex Tél : 04 73 40 50 04 E-mail : alain.quilliot@isima.fr 5