13 research outputs found
MĂ©thodes pour favoriser lâintĂ©gralitĂ© de lâamĂ©lioration dans le simplexe en nombres entiers - Application aux rotations dâĂ©quipages aĂ©riens
RĂSUMĂ : Dans son cadre le plus gĂ©nĂ©ral, le processus dâoptimisation mathĂ©matique se scinde en trois grandes Ă©tapes. La premiĂšre consiste Ă modĂ©liser le problĂšme, câest-Ă -dire le reprĂ©senter sous la forme dâun programme mathĂ©matique, ensemble dâĂ©quations constituĂ© dâun objectif Ă minimiser ou maximiser (typiquement, les coĂ»ts ou le bĂ©nĂ©fice de lâentreprise) et de contraintes Ă satisfaire (contraintes opĂ©rationnelles, convention collective, etc.). Aux dĂ©cisions Ă prendre correspondent les variables du problĂšme. Sâil est une reprĂ©sentation parfaite de la rĂ©alitĂ©, ce modĂšle est dit exact, sinon il reste approximatif. La seconde Ă©tape du processus est la rĂ©solution de ce programme mathĂ©matique. Il sâagit de dĂ©terminer une solution respectant les contraintes et pour laquelle la valeur de lâobjectif est la meilleure possible. Pour ce faire, on applique gĂ©nĂ©ralement un algorithme de rĂ©solution, ensemble de rĂšgles opĂ©ratoires dont lâapplication permet de rĂ©soudre le problĂšme Ă©noncĂ© au moyen dâun nombre fini dâopĂ©rations. Un algorithme peut ĂȘtre traduit grĂące Ă un langage de programmation en un programme exĂ©cutable par un ordinateur. LâexĂ©cution dâun tel programme permet ainsi de rĂ©soudre le programme mathĂ©matique. Enfin, la derniĂšre Ă©tape consiste Ă ajuster la solution obtenue Ă la rĂ©alitĂ©. Dans le cas oĂč le modĂšle nâest quâapproximatif, cette solution peut ne pas convenir
et nĂ©cessiter dâĂȘtre modifiĂ©e a posteriori afin de sâaccorder aux exigences de la rĂ©alitĂ© concrĂšte. Cette thĂšse se concentre sur la seconde de ces trois Ă©tapes, lâĂ©tape de rĂ©solution, en particulier sur le dĂ©veloppement dâun algorithme de rĂ©solution dâun programme mathĂ©matique
prĂ©cis, le partitionnement dâensemble. Le problĂšme de partitionnement dâensemble permet de modĂ©liser des applications variĂ©es : planification dâemplois du temps, logistique, production dâĂ©lectricitĂ©, partage Ă©quitable,
reconnaissance de forme, etc. Pour chacun de ces exemples lâobjectif et les contraintes prennent des significations physiques diffĂ©rentes, mais la structure du modĂšle est la mĂȘme. Dâun point de vue mathĂ©matique, il sâagit dâun programme linĂ©aire en nombres entiers, dont les variables sont binaires, câest-Ă -dire quâelles ne peuvent prendre que les valeurs 0 et 1. Le programme est linĂ©aire car lâobjectif et les contraintes sont reprĂ©sentĂ©s par des fonctions linĂ©aires des variables. Les algorithmes les plus couramment utilisĂ©s pour la rĂ©solution de tels problĂšmes sont basĂ©s sur le principe de sĂ©paration et Ă©valuation (branch-and-bound). Dans ces mĂ©thodes, les contraintes dâintĂ©gralitĂ© sont dâabord relĂąchĂ©es : les solutions peuvent alors ĂȘtre fractionnaires. La rĂ©solution du programme ainsi obtenu â appelĂ© relaxation linĂ©aire du
programme en nombres entiers â est bien plus simple que celle du programme en nombres entiers. Pour obtenir lâintĂ©gralitĂ©, on sĂ©pare le problĂšme afin dâĂ©liminer les solutions fractionnaires. Ces sĂ©parations donnent naissance Ă un arbre de branchement oĂč, Ă chaque noeud, la relaxation dâun problĂšme de partitionnement de la taille du problĂšme original est rĂ©solue. La taille de cet arbre, et donc le temps dâexĂ©cution, croissent exponentiellement avec la taille des instances. De plus, lâalgorithme utilisĂ© pour rĂ©soudre la relaxation, le simplexe, fonctionne mal sur des problĂšmes dĂ©gĂ©nĂ©rĂ©s, câest-Ă -dire dont trop de contraintes sont saturĂ©es. Câest malheureusement le cas de nombreux problĂšmes issus de lâindustrie, particuliĂšrement du problĂšme de partitionnement dont le taux de dĂ©gĂ©nĂ©rescence est intrinsĂšquement Ă©levĂ©. Une autre approche de ce type de problĂšmes est celle des algorithmes primaux : il sâagit
de partir dâune solution entiĂšre non optimale, de trouver une direction qui mĂšne vers une meilleure solution entiĂšre, puis dâitĂ©rer ce processus jusquâĂ atteindre lâoptimalitĂ©. Ă chaque Ă©tape, un sous-problĂšme dâaugmentation est rĂ©solu : trouver une direction dâamĂ©lioration (ou dâaugmentation) ou affirmer que la solution courante est optimale. Les travaux concernant les mĂ©thodes primales sont moins nombreux que ceux sur le branch-and-bound, qui
représentent depuis quarante ans la filiÚre dominante pour la résolution de problÚmes en nombres entiers. Développer une méthode primale efficace en pratique constituerait ainsi
un changement majeur dans le domaine. Des travaux computationels sur des algorithmes primaux ressortent deux principaux dĂ©fis rencontrĂ©s lors de la conception et lâimplĂ©mentation de ces mĂ©thodes. Dâune part, de nombreuses directions dâamĂ©lioration sont irrĂ©alisables, câest-Ă -dire quâeffectuer un pas, aussi petit soit-il, dans ces directions implique une violation des contraintes du problĂšme. On parle alors de dĂ©gĂ©nĂ©rescence ; câest par exemple le cas des directions associĂ©es Ă certains pivots de simplexe (pivots dĂ©gĂ©nĂ©rĂ©s). Les directions irrĂ©alisables
ne permettent pas Ă lâalgorithme de progresser et peuvent mettre en pĂ©ril sa terminaison sâil est impossible de dĂ©terminer de direction rĂ©alisable. Dâautre part, lorsquâune
direction dâamĂ©lioration rĂ©alisable pour la relaxation linĂ©aire a Ă©tĂ© dĂ©terminĂ©e, il est difficile de sâassurer que la solution vers laquelle elle mĂšne est entiĂšre. Parmi les algorithmes primaux existants, celui qui apparait comme le plus prometteur est le simplexe en nombres entiers
avec dĂ©composition (Integral Simplex Using Decomposition, ISUD) car il intĂšgre au cadre primal des techniques de dĂ©composition permettant de se prĂ©munir des effets nĂ©fastes de la dĂ©gĂ©nĂ©rescence. Il sâagit Ă notre connaissance du premier algorithme de type primal capable de battre le branch-and-bound sur des instances de grande taille ; par ailleurs, la diffĂ©rence est dâautant plus importante que le problĂšme est grand. Bien que fournissant des Ă©lĂ©ments
de rĂ©ponse Ă la problĂ©matique de la dĂ©gĂ©nĂ©rescence, cette mĂ©thode nâaborde pas pour autant la question de lâintĂ©gralitĂ© lors du passage Ă une solution de meilleur coĂ»t ; et pour quâISUD puisse envisager de supplanter les mĂ©thodes de type branch-and-bound, il lui faut parcourir
cette deuxiĂšme moitiĂ© du chemin. Il sâagit lĂ de lâobjectif de ce doctorat : augmenter le taux de directions entiĂšres trouvĂ©es par ISUD pour le rendre applicable aux instances industrielles de grande taille, de type planification de personnel. Pour aller dans cette direction, nous approfondissons tout dâabord les connaissances thĂ©oriques
sur ISUD. Formuler ce dernier comme un algorithme primal, comprendre en quoi il se rattache à cette famille, le traduire pour la premiÚre fois dans un langage exclusivement primal sans faire appel à la dualité, constituent le terreau de cette thÚse. Cette analyse permet
ensuite de mieux dĂ©crire la gĂ©omĂ©trie sous-jacente ainsi que les domaines de rĂ©alisabilitĂ© des diffĂ©rents problĂšmes linĂ©aires considĂ©rĂ©s. Quand bien mĂȘme ce pan majeur de notre travail nâest pas prĂ©sentĂ© dans cette thĂšse comme un chapitre Ă part entiĂšre, il se situe indubitablement
Ă lâorigine de chacune de nos idĂ©es, de nos approches et de nos contributions. Cette approche de lâalgorithme sous un angle nouveau donne lieu Ă de nombreuses simplifications,
améliorations et extensions de résultats déjà connus.
Dans un premier temps, nous gĂ©nĂ©ralisons la formulation du problĂšme dâaugmentation afin dâaugmenter la probabilitĂ© que la direction dĂ©terminĂ©e par lâalgorithme mĂšne vers une nouvelle solution entiĂšre. Lors de lâexĂ©cution dâISUD, pour dĂ©terminer la direction qui mĂšnera Ă la solution suivante, on rĂ©sout un programme linĂ©aire dont la solution est une direction dâamĂ©lioration qui appartient au cĂŽne des directions rĂ©alisables. Pour sâassurer que ce programme
est bornĂ© (les directions pourraient partir Ă lâinfini), on lui ajoute une contrainte de normalisation et on se restreint ainsi Ă une section de ce cĂŽne. Dans la version originelle de lâalgorithme, les coefficients de cette contrainte sont uniformes. Nous gĂ©nĂ©ralisons cette contrainte Ă une section quelconque du cĂŽne et montrons que la direction rĂ©alisable dĂ©terminĂ©e par lâalgorithme dĂ©pend fortement du choix des coefficients de cette contrainte ;
il en va de mĂȘme pour la probabilitĂ© que la solution vers laquelle elle mĂšne soit entiĂšre. Nous Ă©tendons les propriĂ©tĂ©s thĂ©oriques liĂ©s Ă la dĂ©composition dans lâalgorihtme ISUD et montrons de nouveaux rĂ©sultats dans le cas dâun choix de coefficients quelconques. Nous dĂ©terminons
de nouvelles propriétés spécifiques à certains choix de normalisation et faisons des recommandations pour choisir les coefficients afin de pénaliser les directions fractionnaires au profit des directions entiÚres. Des résultats numériques sur des instances de planification
de personnel montrent le potentiel de notre approche. Alors que la version originale dâISUD permet de rĂ©soudre 78% des instances de transport aĂ©rien du benchmark considĂ©rĂ©, 100%
sont rĂ©solues grĂące Ă lâun, au moins, des modĂšles que nous proposons. Dans un second temps, nous montrons quâil est possible dâadapter des mĂ©thodes de plans coupants utilisĂ©s en programmation linĂ©aire en nombres entiers au cas dâISUD. Nous montrons que des coupes peuvent ĂȘtres transfĂ©rĂ©es dans le problĂšme dâaugmentation, et nous caractĂ©risons lâensemble des coupes transfĂ©rables comme lâensemble, non vide, des coupes primales saturĂ©es pour la solution courante du problĂšme de partitionnement. Nous montrons
que de telles coupes existent toujours, proposons des algorithmes de sĂ©paration efficaces pour les coupes primales de cycle impair et de clique, et montrons que lâespace de recherche de ces coupes peut ĂȘtre restreint Ă un petit nombre de variables, ce qui rend le processus efficace. Des rĂ©sultats numĂ©riques prouvent la validitĂ© de notre approche ; ces tests sont effectuĂ©s sur des instances de planification de personnel navigant et de chauffeurs dâautobus allant jusquâĂ 1 600 contraintes et 570 000 variables. Sur les instances de transport aĂ©rien testĂ©es
lâajout de coupes primales permet de passer dâun taux de rĂ©solution de 70% Ă 92%. Sur de grandes instances dâhoraires de chauffeurs dâautobus, les coupes prouvent lâoptimalitĂ© locale de la solution dans plus de 80% des cas.
Dans un dernier temps, nous modifions dynamiquement les coefficients de la contrainte de normalisation lorsque la direction trouvĂ©e par lâalgorithme mĂšne vers une solution fractionnaire. Nous proposons plusieurs stratĂ©gies de mise-Ă -jour visant Ă pĂ©naliser les directions fractionnaires basĂ©es sur des observations thĂ©oriques et pratiques. Certaines visent Ă pĂ©naliser la direction choisie par lâalgorithme, dâautres procĂšdent par perturbation des coefficients de normalisation en utilisant les Ă©quations des coupes mentionnĂ©es prĂ©cĂ©demment. Cette version de lâalgorithme est testĂ©e sur un nouvel ensemble dâinstances provenant de lâindustrie du transport aĂ©rien. Ă notre connaissance, lâensemble dâinstances que nous proposons
nâest comparable Ă aucun autre. Il sâagit en effet de grands problĂšmes dâhoraires de personnel navigant allant jusquâĂ 1 700 vols et 115 000 rotations, donc autant de contraintes et de variables. Ils sont posĂ©s sous la forme de problĂšmes de partitionnement pour lesquels nous fournissons des solutions initiales comparables Ă celles dont on disposerait en milieu industriel. Notre travail montre le potentiel quâont les algorithmes primaux pour rĂ©soudre des problĂšmes de planification de personnel navigant, problĂšmes clĂ©s pour les compagnies aĂ©riennes, tant par leur complexitĂ© intrinsĂšque que par les consĂ©quences Ă©conomiques et financiĂšres quâils entraĂźnent.----------ABSTRACT : Optimization is a three-step process. Step one models the problem and writes it as a mathematical program, i.e., a set of equations that includes an objective one seeks to minimize or maximize (typically the costs or benefit of a company) and constraints that must be satisfied by any acceptable solution (operational constraints, collective agreement, etc.). The unknowns of the model are the decision variables; they correspond to the quantities the
decision-maker wants to infer. A model that perfectly represents reality is exact, otherwise it is approximate. The second step of the optimization process is the solution of the mathematical program, i.e., the determination of a solution that satisfies all constraints and for which the objective value is as good as possible. To this end, one generally uses an algorithm, a self-contained step-by-step set of operating rules that solves the problem in a finite
number of operations. The algorithm is translated by means of a programming language into an executable program run by a computer; the execution of such software solves the
mathematical program. Finally, the last step is the adaptation of the mathematical solution to reality. When the model is only approximate, the output solution may not fit the original requirements and therefore require a posteriori modifications. This thesis concentrates
on the second of these three steps, the solution process. More specifically, we design and implement an algorithm that solves a specific mathematical program: set partitioning. The set partitioning problem models a very wide range of applications: workforce scheduling,
logistics, electricity production planning, pattern recognition, etc. In each of these examples, the objective function and the constraints have different physical significations but the structure of the model is the same. From a mathematical point of view, it is an integer
linear program whose decision variables can only take value 0 or 1. It is linear because both the objective and the constraints are linear functions of the variables. Most algorithms used to solve this family of programs are based on the principle called branch-and-bound. At first, the integrality constraints are relaxed; solutions may thus be fractional. The solution of the resulting program â called linear relaxation of the integer program â is significantly
easier than that of the integer program. Then, to recover integrality, the problem is separated to eliminate fractional solutions. From the splitting a branching tree arises, in which, at each node, the relaxation of a set partitioning problem as big as the original one is solved.
The size of that tree, and thus the solving time, grows exponentially with the size of the instance. Furthermore, the algorithm that solves the linear relaxations, the simplex, performs poorly on degenerate problems, i.e., problems for which too many constraints are tight. It is
unfortunately the case of many industrial problems, and particularly of the set partitioning problem whose degeneracy rate is intrinsiquely high. An alternative approach is that of primal algorithms: start from a nonoptimal integer solution and find a direction that leads to a better one (also integer). That process is iterated
until optimality is reached. At each step of the process one solves an augmentation subproblem which either outputs an augmenting direction or asserts that the current solution is optimal. The literature is significantly less abundant on primal algorithms than on branchand-
bound and the latter has been the dominant method in integer programming for over forty years. The development of an efficient primal method would therefore stand as a major breakthrough in this field. From the computational works on primal algorithms, two main issues stand out concerning their design and implementation. On the one hand, many augmenting directions are infeasible, i.e., taking the smallest step in such a direction results
in a violation of the constraints. This problem is strongly related to degeneracy and often affects simplex pivots (e.g., degenerate pivots). Infeasible directions prevent the algorithm from moving ahead and may jeopardize its performance, and even its termination when it is impossible to find a feasible direction. On the other hand, when a cost-improving direction has been succesfully determined, it may be hard to ensure that it leads to an integer solution. Among existing primal algorithms, the one appearing to be the most promising is the integral simplex using decomposition (ISUD) because it embeds decomposition techniques that palliate the unwanted effects of degeneracy into a primal framework. To our knowledge, it is the first primal algorithm to beat branch-and-bound on large scale industrial instances. Furthermore, its performances improve when the problem gets bigger. Despite its strong assets
to counter degeneracy, however, this method does not handle the matter of integrality when reaching out for the next solution; and if ISUD is to compete with branch-and-bound,
it is crucial that this issue be tackled. Therefore, the purpose of this thesis is the following: increasing the rate of integral directions found by ISUD to make it fully
competitive with existing solvers on large-scale industrial workforce scheduling instances. To proceed in that direction, we first deepen the theoretical knowledge on ISUD. Formulating it as a primal algorithm, understanding how it belongs to that family, and translating it in a purely primal language that requires no notion of duality provide a fertile ground to our work. This analysis yields geometrical interpretations of the underlying structures and
domains of the several mathematical programs involved in the solution process. Although no chapter specifically focuses on that facet of our work, most of our ideas, approaches and contributions stem from it. This groundbreaking approach of ISUD leads to simplifications,
strengthening, and extensions of several theoretical results. In the first part of this work, we generalize the formulation of the augmentation problem in order to increase the likelihood that the direction found by the algorithm leads to a new integer solution. In ISUD, to find the edge leading to the next point, one solves a linear
program to select an augmenting direction from a cone of feasible directions. To ensure that this linear program is bounded (the directions could go to infinity), a normalization constraint is added and the optimization is performed on a section of the cone. In the original version of the algorithm, all weights take the same value. We extend this constraint to the case of a generic normalization constraint and show that the output direction dépends strongly on the chosen normalization weights, and so does the likelihood that the next solution is integer. We extend the theoretical properties of ISUD, particularly those that are related to decomposition and we prove new results in the case of a generic normalization constraint. We explore the theoretical properties of some specific constraints, and discuss the design of the normalization constraint so as to penalize fractional directions. We also
report computational results on workforce scheduling instances that show the potential behind our approach. While only 78% of aircrew scheduling instances from that benchmark are solved with the original version of ISUD, 100% of them are solved by at least one of the models we propose. In the second part, we show that cutting plane methods used in integer linear programming can be adapted to ISUD. We show that cutting planes can be transferred to the augmentation problem, and we characterize the set of transferable cuts as a nonempty subset of primal cuts that are tight to the current solution. We prove that these cutting planes always exist, we propose efficient separation procedures for primal clique and odd-cycle cuts, and we prove that their search space can be restricted to a small subset of the variables making the computation efficient. Numerical results demonstrate the effectiveness of adding cutting planes to the algorithm. Tests are performed on small- and large-scale set partitioning problems from aircrew and bus-driver scheduling instances up to 1,600 constraints and 570,000 variables. On the aircrew scheduling instances, the addition of primal cuts raises the rate of instances solved from 70% to 92%. On large bus drivers scheduling instances, primal cuts prove that the solution found by ISUD is optimal over a large subset of the domain for more than 80% of the instances.
In the last part, we dynamically update the coefficients of the normalization constraint whenever the direction found by the algorithm leads to a fractional solution, to penalize
that direction. We propose several update strategies based on theoretical and experimental results. Some penalize the very direction returned by the algorithm, others operate by
perturbating the normalization coefficients with those of the aforementionned primal cuts. To prove the efficiency of our strategies, we show that our version of the algorithm yields better results than the former version and than classical branch-and-bound techniques on a benchmark of industrial aircrew scheduling instances. The benchmark that we propose is, to the best of our knowledge, comparable to no other from the literature. It provides largescale
instances with up to 1,700 flights and 115,000 pairings, hence as many constraints and variables, and the instances are given in a set-partitioning form together with initial
solutions that accurately mimic those of industrial applications. Our work shows the strong potential of primal algorithms for the crew scheduling problem, which is a key challenge for large airlines, both financially significant and notably hard to solve
Improved Primal Simplex: A More General Theoretical Framework and an Extended Experimental Analysis
International audienceIn this article, we propose a general framework for an algorithm derived from the primal simplex that guarantees a strict improvement in the objective after each iteration. Our approach relies on the identification of compatible variables that ensure a nondegenerate iteration if pivoted into the basis. The problem of finding a strict improvement in the objective function is proved to be equivalent to two smaller problems respectively focusing on compatible and incompatible variables. We then show that the improved primal simplex (IPS) of Elhallaoui et al. is a particular implementation of this generic theoretical framework. The resulting new description of IPS naturally emphasizes what should be considered as necessary adaptations of the framework versus specific implementation choices. This provides original insight into IPS that allows for the identification of weaknesses and potential alternative choices that would extend the efficiency of the method to a wider set of problems. We perform experimental tests on an extended collection of data sets including instances of Mittelmann's benchmark for linear programming. The results confirm the excellent potential of IPS and highlight some of its limits while showing a path toward an improved implementation of the generic algorithm
Evaluation of appendicitis risk prediction models in adults with suspected appendicitis
Background
Appendicitis is the most common general surgical emergency worldwide, but its diagnosis remains challenging. The aim of this study was to determine whether existing risk prediction models can reliably identify patients presenting to hospital in the UK with acute right iliac fossa (RIF) pain who are at low risk of appendicitis.
Methods
A systematic search was completed to identify all existing appendicitis risk prediction models. Models were validated using UK data from an international prospective cohort study that captured consecutive patients aged 16â45âyears presenting to hospital with acute RIF in March to June 2017. The main outcome was best achievable model specificity (proportion of patients who did not have appendicitis correctly classified as low risk) whilst maintaining a failure rate below 5 per cent (proportion of patients identified as low risk who actually had appendicitis).
Results
Some 5345 patients across 154 UK hospitals were identified, of which twoâthirds (3613 of 5345, 67·6 per cent) were women. Women were more than twice as likely to undergo surgery with removal of a histologically normal appendix (272 of 964, 28·2 per cent) than men (120 of 993, 12·1 per cent) (relative risk 2·33, 95 per cent c.i. 1·92 to 2·84; Pâ<â0·001). Of 15 validated risk prediction models, the Adult Appendicitis Score performed best (cutâoff score 8 or less, specificity 63·1 per cent, failure rate 3·7 per cent). The Appendicitis Inflammatory Response Score performed best for men (cutâoff score 2 or less, specificity 24·7 per cent, failure rate 2·4 per cent).
Conclusion
Women in the UK had a disproportionate risk of admission without surgical intervention and had high rates of normal appendicectomy. Risk prediction models to support shared decisionâmaking by identifying adults in the UK at low risk of appendicitis were identified
An Online Stochastic Algorithm for a Dynamic Nurse Scheduling Problem
International audienceIn this paper, we focus on the problem studied in the second international nurse rostering competition: a personalized nurse scheduling problem under uncertainty. The schedules must be computed week by week over a planning horizon of up to eight weeks. We present the work that the authors submitted to this competition and which was awarded the second prize. At each stage, the dynamic algorithm is fed with the staffing demand and nurses preferences for the current week and computes an irrevocable schedule for all nurses without knowledge of future inputs. The challenge is to obtain a feasible and near-optimal schedule at the end of the horizon. The online stochastic algorithm described in this paper draws inspiration from the primal-dual algorithm for online optimization and the sample average approximation, and is built upon an existing static nurse scheduling software. The procedure generates a small set of candidate schedules, rank them according to their performance over a set of test scenarios, and keeps the best one. Numerical results show that this algorithm is very robust, since it has been able to produce feasible and near optimal solutions on most of the proposed instances ranging from 30 to 120 nurses over a horizon of 4 or 8 weeks. Finally, the code of our implementation is open source and available in a public repository
A rotation-based branch-and-price approach for the nurse scheduling problem
International audienceIn this paper, we describe an algorithm for the personalized nurse scheduling problem. We focus on the deterministic counterpart of the specific problem that has been described in the second international nurse rostering competition. One specificity of this version of the problem is that most constraints are soft, meaning that they can be violated at the price of a penalty. The feasible space is thus much larger, which involves much more difficulty to find the optimal solution. We model the problem as a an integer program (IP) that we solve using a branch-and-price procedure. This model is, to the best of our knowledge, comparable to no other from the literature, since each column of the IP corresponds to a rotation, i.e., a sequence of consecutive worked days for a nurse, and not to a complete individual roster. We tackle instances involving up to 120 nurses and 4 shifts over an 8-weeks horizon by embedding the branch-and-price in a large-neighborhood-search framework. Initial solutions of the large-neighborhood search are found by a rolling-horizon algorithm, well-suited to the rotation model