3.1 Introduction

Keywords: integer programming, graph theory, decomposition approaches, polyhedral approaches,quadratic programming approaches, constraint programming..

Combinatorial optimization is the field of discrete optimization problems. In many applications, the most important decisions (control variables) are binary (on/off decisions) or integer (indivisible quantities). Extra variables can represent continuous adjustments or amounts. This results in models known as mixed integer programs (MIP), where the relationships between variables and input parameters are expressed as linear constraints and the goal is defined as a linear objective function. MIPs are notoriously difficult to solve: good quality estimations of the optimal value (bounds) are required to prune enumeration-based global-optimization algorithms whose complexity is exponential. In the standard approach to solving an MIP is so-called branch-and-bound algorithm : $\left(i\right)$ one solves the linear programming (LP) relaxation using the simplex method; $\left(ii\right)$ if the LP solution is not integer, one adds a disjunctive constraint on a factional component (rounding it up or down) that defines two sub-problems; $\left(iii\right)$ one applies this procedure recursively, thus defining a binary enumeration tree that can be pruned by comparing the local LP bound to the best known integer solution. Commercial MIP solvers are essentially based on branch-and-bound (such IBM-CPLEX, FICO-Xpress-mp, or GUROBI). They have made tremendous progress over the last decade (with a speedup by a factor of 60). But extending their capabilities remains a continuous challenge; given the combinatorial explosion inherent to enumerative solution techniques, they remain quickly overwhelmed beyond a certain problem size or complexity.

Progress can be expected from the development of tighter formulations. Central to our field is the characterization of polyhedra defining or approximating the solution set and combinatorial algorithms to identify “efficiently” a minimum cost solution or separate an unfeasible point. With properly chosen formulations, exact optimization tools can be competitive with other methods (such as meta-heuristics) in constructing good approximate solutions within limited computational time, and of course has the important advantage of being able to provide a performance guarantee through the relaxation bounds. Decomposition techniques are implicitly leading to better problem formulation as well, while constraint propagation are tools from artificial intelligence to further improve formulation through intensive preprocessing. A new trend is robust optimization where recent progress have been made: the aim is to produce optimized solutions that remain of good quality even if the problem data has stochastic variations. In all cases, the study of specific models and challenging industrial applications is quite relevant because developments made into a specific context can become generic tools over time and see their way into commercial software.

Our project brings together researchers with expertise in mathematical programming (polyhedral approaches, decomposition and reformulation techniques in mixed integer programing, robust and stochastic programming, and dynamic programming), graph theory (characterization of graph properties, combinatorial algorithms) and constraint programming in the aim of producing better quality formulations and developing new methods to exploit these formulations. These new results are then applied to find high quality solutions for practical combinatorial problems such as routing, network design, planning, scheduling, cutting and packing problems, High Performance and Cloud Computing.

3.2 Polyhedral approaches for MIP

Adding valid inequalities to the polyhedral description of an MIP allows one to improve the resulting LP bound and hence to better prune the enumeration tree. In a cutting plane procedure, one attempt to identify valid inequalities that are violated by the LP solution of the current formulation and adds them to the formulation. This can be done at each node of the branch-and-bound tree giving rise to a so-called branch-and-cut algorithm45. The goal is to reduce the resolution of an integer program to that of a linear program by deriving a linear description of the convex hull of the feasible solutions. Polyhedral theory tells us that if $X$ is a mixed integer program: $X=P\cap {\mathbb{Z}}^n\times {ℝ}^p$ where $P=\{x\in {ℝ}^{n+p}:Ax\le b\}$ with matrix $(A,b)\in {\mathbb{Q}}^{m\times (n+p+1)}$, then $conv\left(X\right)$ is a polyhedron that can be described in terms of linear constraints, i.e. it writes as $conv\left(X\right)=\{x\in {ℝ}^{n+p}:Cx\le d\}$ for some matrix $(C,d)\in {\mathbb{Q}}^{m^{\text{'}}\times (n+p+1)}$ although the dimension $m^{\text{'}}$ is typically quite large. A fundamental result in this field is the equivalence of complexity between solving the combinatorial optimization problem $min\{cx:x\in X\}$ and solving the separation problem over the associated polyhedron $conv\left(X\right)$: if $\tilde{x}\notin conv\left(X\right)$, find a linear inequality $\pi x\ge {\pi }_0$ satisfied by all points in $conv\left(X\right)$ but violated by $\tilde{x}$. Hence, for NP-hard problems, one can not hope to get a compact description of $conv\left(X\right)$ nor a polynomial time exact separation routine. Polyhedral studies focus on identifying some of the inequalities that are involved in the polyhedral description of $conv\left(X\right)$ and derive efficient separation procedures (cutting plane generation). Only a subset of the inequalities $Cx\le d$ can offer a good approximation, that combined with a branch-and-bound enumeration techniques permits to solve the problem. Using cutting plane algorithm at each node of the branch-and-bound tree, gives rise to the algorithm called branch-and-cut.

3.3 Decomposition-and-reformulation-approaches

An hierarchical approach to tackle complex combinatorial problems consists in considering separately different substructures (subproblems). If one is able to implement relatively efficient optimization on the substructures, this can be exploited to reformulate the global problem as a selection of specific subproblem solutions that together form a global solution. If the subproblems correspond to subset of constraints in the MIP formulation, this leads to Dantzig-Wolfe decomposition. If it corresponds to isolating a subset of decision variables, this leads to Bender's decomposition. Both lead to extended formulations of the problem with either a huge number of variables or constraints. Dantzig-Wolfe approach requires specific algorithmic approaches to generate subproblem solutions and associated global decision variables dynamically in the course of the optimization. This procedure is known as column generation, while its combination with branch-and-bound enumeration is called branch-and-price. Alternatively, in Bender's approach, when dealing with exponentially many constraints in the reformulation, the cutting plane procedures that we defined in the previous section are well-suited tools. When optimization on a substructure is (relatively) easy, there often exists a tight reformulation of this substructure typically in an extended variable space. This gives rise powerful reformulation of the global problem, although it might be impractical given its size (typically pseudo-polynomial). It can be possible to project (part of) the extended formulation in a smaller dimensional space if not the original variable space to bring polyhedral insight (cuts derived through polyhedral studies can often be recovered through such projections).

3.4 Integration of Artificial Intelligence Techniques in Integer Programming

When one deals with combinatorial problems with a large number of integer variables, or tightly constrained problems, mixed integer programming (MIP) alone may not be able to find solutions in a reasonable amount of time. In this case, techniques from artificial intelligence can be used to improve these methods. In particular, we use variable fixing techniques, primal heuristics and constraint programming.

Primal heuristics are useful to find feasible solutions in a small amount of time. We focus on heuristics that are either based on integer programming (rounding, diving, relaxation induced neighborhood search, feasibility pump), or that are used inside our exact methods (heuristics for separation or pricing subproblem, heuristic constraint propagation, ...). Such methods are likely to produce good quality solutions only if the integer programming formulation is of top quality, i.e., if its LP relaxation provides a good approximation of the IP solution.

In the same line, variable fixing techniques, that are essential in reducing the size of large scale problems, rely on good quality approximations: either tight formulations or tight relaxation solvers (as a dynamic program combined with state space relaxation). Then if the dual bound derives when the variable is fixed to one exceeds the incubent solution value, the variable can be fixed to zero and hence removed from the problem. The process can be apply sequentially by refining the degree of relaxation.

Constraint Programming (CP) focuses on iteratively reducing the variable domains (sets of feasible values) by applying logical and problem-specific operators. The latter propagates on selected variables the restrictions that are implied by the other variable domains through the relations between variables that are defined by the constraints of the problem. Combined with enumeration, it gives rise to exact optimization algorithms. A CP approach is particularly effective for tightly constrained problems, feasibility problems and min-max problems. Mixed Integer Programming (MIP), on the other hand, is known to be effective for loosely constrained problems and for problems with an objective function defined as the weighted sum of variables. Many problems belong to the intersection of these two classes. For such problems, it is reasonable to use algorithms that exploit complementary strengths of Constraint Programming and Mixed Integer Programming.

3.5 Robust Optimization

Decision makers are usually facing several sources of uncertainty, such as the variability in time or estimation errors. A simplistic way to handle these uncertainties is to overestimate the unknown parameters. However, this results in over-conservatism and a significant waste in resource consumption. A better approach is to account for the uncertainty directly into the decision aid model by considering mixed integer programs that involve uncertain parameters. Stochastic optimization account for the expected realization of random data and optimize an expected value representing the average situation. Robust optimization on the other hand entails protecting against the worst-case behavior of unknown data. There is an analogy to game theory where one considers an oblivious adversary choosing the realization that harms the solution the most. A full worst case protection against uncertainty is too conservative and induces very high over-cost. Instead, the realization of random data are bound to belong to a restricted feasibility set, the so-called uncertainty set. Stochastic and robust optimization rely on very large scale programs where probabilistic scenarios are enumerated. There is hope of a tractable solution for realistic size problems, provided one develops very efficient ad-hoc algorithms. The techniques for dynamically handling variables and constraints (column-and-row generation and Bender's projection tools) that are at the core of our team methodological work are specially well-suited to this context.

3.6 Polyhedral Combinatorics and Graph Theory

Many fundamental combinatorial optimization problems can be modeled as the search for a specific structure in a graph. For example, ensuring connectivity in a network amounts to building a tree that spans all the nodes. Inquiring about its resistance to failure amounts to searching for a minimum cardinality cut that partitions the graph. Selecting disjoint pairs of objects is represented by a so-called matching. Disjunctive choices can be modeled by edges in a so-called conflict graph where one searches for stable sets – a set of nodes that are not incident to one another. Polyhedral combinatorics is the study of combinatorial algorithms involving polyhedral considerations. Not only it leads to efficient algorithms, but also, conversely, efficient algorithms often imply polyhedral characterizations and related min-max relations. Developments of polyhedral properties of a fundamental problem will typically provide us with more interesting inequalities well suited for a branch-and-cut algorithm to more general problems. Furthermore, one can use the fundamental problems as new building bricks to decompose the more general problem at hand. For problem that let themselves easily be formulated in a graph setting, the graph theory and in particular graph decomposition theorem might help.

4.1 Network Design and Routing Problems

We are actively working on problems arising in network topology design, implementing a survivability condition of the form “at least two paths link each pair of terminals”. We have extended polyhedral approaches to problem variants with bounded length requirements and re-routing restrictions 43. Associated to network design is the question of traffic routing in the network: one needs to check that the network capacity suffices to carry the demand for traffic. The assignment of traffic also implies the installation of specific hardware at transient or terminal nodes.

To accommodate the increase of traffic in telecommunication networks, today's optical networks use grooming and wavelength division multiplexing technologies. Packing multiple requests together in the same optical stream requires to convert the signal in the electrical domain at each aggregation of disaggregation of traffic at an origin, a destination or a bifurcation node. Traffic grooming and routing decisions along with wavelength assignments must be optimized to reduce opto-electronics system installation cost. We developed and compared several decomposition approaches 59 to deal with backbone optical network with relatively few nodes (around 20) but thousands of requests for which traditional multi-commodity network flow approaches are completely overwhelmed. We also studied the impact of imposing a restriction on the number of optical hops in any request route. We also developed a branch-and-cut approach to a problem that consists in placing sensors on the links of a network for a minimum cost 46.

The Dial-a-Ride Problem is a variant of the pickup and delivery problem with time windows, where the user inconvenience must be taken into account. In  52, ride time and customer waiting time are modeled through both constraints and an associated penalty in the objective function. We develop a column generation approach, dynamically generating feasible vehicle routes. Handling ride time constraints explicitly in the pricing problem solver requires specific developments. Our dynamic programming approach for pricing problem makes use of a heuristic dominance rule and a heuristic enumeration procedure, which in turns implies that our overall branch-and-price procedure is a heuristic. However, in practice our heuristic solutions are experimentally very close to exact solutions and our approach is numerically competitive in terms of computation times.

In  49, 50, we consider the problem of covering an urban area with sectors under additional constraints. We adapt the aggregation method to our column generation algorithm and focus on the problem of disaggregating the dual solution returned by the aggregated master problem.

We studied several time dependent formulations for the unit demand vehicle routing problem  34, 35. We gave new bounding flow inequalities for a single commodity flow formulation of the problem. We described their impact by projecting them on some other sets of variables, such as variables issued of the Picard and Queyranne formulation or the natural set of design variables. Some inequalities obtained by projection are facet defining for the polytope associated with the problem. We are now running more numerical experiments in order to validate in practice the efficiency of our theoretical results.

We considered the multi-commodity transportation problem. Applications of this problem arise in, for example, rail freight service design, "less than truckload" trucking, where goods should be delivered between different locations in a transportation network using various kinds of vehicles of large capacity. A particularity here is that, to be profitable, transportation of goods should be consolidated. This means that goods are not delivered directly from the origin to the destination, but transferred from one vehicle to another in intermediate locations. We proposed an original Mixed Integer Programming formulation for this problem which is suitable for resolution by a Branch-and-Price algorithm and intelligent primal heuristics based on it.

For the problem of routing freight railcars, we proposed two algorithmes based on the column generation approach. These algorithmes have been tested on a set of real-life instances coming from a real Russian freight transportation company. Our algorithms have been faster on these instances than the current solution approach being used by the company.

4.2 Packing and Covering Problems

Realopt team has a strong experience on exact methods for cutting and packing problems. These problems occur in logistics (loading trucks), industry (wood or steel cutting), computer science (parallel processor scheduling).

We developed a branch-and-price algorithm for the Bin Packing Problem with Conflicts which improves on other approaches available in the literature 57. The algorithm uses our methodological advances like the generic branching rule for the branch-and-price and the column based heuristic. One of the ingredients which contributes to the success of our method are fast algorithms we developed for solving the subproblem which is the Knapsack Problem with Conflicts. Two variants of the subproblem have been considered: with interval and arbitrary conflict graphs.

We also developed a branch-and-price algorithm for a variant of the bin-packing problem where the items are fragile. In 27 we studied empirically different branching schemes and different algorithms for solving the subproblems.

We studied a variant of the knapsack problem encountered in inventory routing problem: we faced a multiple-class integer knapsack problem with setups 47 (items are partitioned into classes whose use implies a setup cost and associated capacity consumption). We showed the extent to which classical results for the knapsack problem can be generalized to this variant with setups and we developed a specialized branch-and-bound algorithm.

We studied the orthogonal knapsack problem, with the help of graph theory. Fekete and Schepers proposed to model multi-dimensional orthogonal placement problems by using an efficient representation of all geometrically symmetric solutions by a so called packing class involving one interval graph for each dimension. Though Fekete & Schepers' framework is very efficient, we have however identified several weaknesses in their algorithms: the most obvious one is that they do not take advantage of the different possibilities to represent interval graphs. We propose to represent these graphs by matrices with consecutive ones on each row. We proposed a branch-and-bound algorithm for the 2D knapsack problem that uses our 2D packing feasibility check. We are currently developing exact optimization tools for glass-cutting problems in a collaboration with Saint-Gobain 30. This 2D-3stage-Guillotine cut problems are very hard to solve given the scale of the instance we have to deal with. Moreover one has to issue cutting patterns that avoid the defaults that are present in the glass sheet that are used as raw material. There are extra sequencing constraints regarding the production that make the problem even more complex.

We have also organized a European challenge on packing with society Renault. This challenge was about loading trucks under practical constraints.

4.3 Planning, Scheduling, and Logistic Problems

Inventory routing problems combine the optimization of product deliveries (or pickups) with inventory control at customer sites. We considered an industrial application where one must construct the planning of single product pickups over time; each site accumulates stock at a deterministic rate; the stock is emptied on each visit. We have developed a branch-and-price algorithm where periodic plans are generated for vehicles by solving a multiple choice knapsack subproblem, and the global planning of customer visits is coordinated by the master program  48. We previously developed approximate solutions to a related problem combining vehicle routing and planning over a fixed time horizon (solving instances involving up to 6000 pick-ups and deliveries to plan over a twenty day time horizon with specific requirements on the frequency of visits to customers 44.

Together with our partner company GAPSO from the associate team SAMBA, we worked on the equipment routing task scheduling problem  51 arising during port operations. In this problem, a set of tasks needs to be performed using equipments of different types with the objective to maximize the weighted sum of performed tasks.

We participated to the project on an airborne radar scheduling. For this problem, we developed fast heuristics  42 and exact algorithms  29. A substantial research has been done on machine scheduling problems. A new compact MIP formulation was proposed for a large class of these problems 28. An exact decomposition algorithm was developed for the NP-hard maximizing the weighted number of late jobs problem on a single machine  53. A dominant class of schedules for malleable parallel jobs was discovered in the NP-hard problem to minimize the total weighted completion time  55. We proved that a special case of the scheduling problem at cross docking terminals to minimize the storage cost is polynomially solvable  56, 54.

Another application area in which we have successfully developed MIP approaches is in the area of tactical production and supply chain planning. In 26, we proposed a simple heuristic for challenging multi-echelon problems that makes effective use of a standard MIP solver. 25 contains a detailed investigation of what makes solving the MIP formulations of such problems challenging; it provides a survey of the known methods for strengthening formulations for these applications, and it also pinpoints the specific substructure that seems to cause the bottleneck in solving these models. Finally, the results of 31 provide demonstrably stronger formulations for some problem classes than any previously proposed. We are now working on planning phytosanitary treatments in vineries.

We have been developing robust optimization models and methods to deal with a number of applications like the above in which uncertainty is involved. In 40, 39, we analyzed fundamental MIP models that incorporate uncertainty and we have exploited the structure of the stochastic formulation of the problems in order to derive algorithms and strong formulations for these and related problems. These results appear to be the first of their kind for structured stochastic MIP models. In addition, we have engaged in successful research to apply concepts such as these to health care logistics 32. We considered train timetabling problems and their re-optimization after a perturbation in the network 60, 58. The question of formulation is central. Models of the literature are not satisfactory: continuous time formulations have poor quality due to the presence of discrete decision (re-sequencing or re-routing); arc flow in time-space graph blow-up in size (they can only handle a single line timetabling problem). We have developed a discrete time formulation that strikes a compromise between these two previous models. Based on various time and network aggregation strategies, we develop a 2-stage approach, solving the contiguous time model having fixed the precedence based on a solution to the discrete time model.

Currently, we are conducting investigations on a real-world planning problem in the domain of energy production, in the context of a collaboration with EDF  37, 36, 38. The problem consists in scheduling maintenance periods of nuclear power plants as well as production levels of both nuclear and conventional power plants in order to meet a power demand, so as to minimize the total production cost. For this application, we used a Dantzig-Wolfe reformulation which allows us to solve realistic instances of the deterministic version of the problem 41. In practice, the input data comprises a number of uncertain parameters. We deal with a scenario-based stochastic demand with help of a Benders decomposition method. We are working on Multistage Robust Optimization approaches to take into account other uncertain parameters like the duration of each maintenance period, in a dynamic optimization framework. The main challenge addressed in this work is the joint management of different reformulations and solving techniques coming from the deterministic (Dantzig-Wolfe decomposition, due to the large scale nature of the problem), stochastic (Benders decomposition, due to the number of demand scenarios) and robust (reformulations based on duality and/or column and/or row generation due to maintenance extension scenarios) components of the problem 33.

5.1 Footprint of research activities

Our research involves a large amount of computational experiments.

5.2 Impact of research results

The objective of our research is to reduce the quantity of energy/material used to realize some large projects, including energy production and distribution, chemical treatments, and distribution of goods.

7.1 New software

8.1 Algorithms for optimization under uncertainty

In 21, we introduce a new exact algorithm to solve two-stage stochastic linear programs. Based on the multicut Benders reformulation of such problems, with one subproblem for each scenario, this method relies on a partition of the subproblems into batches. By detecting as soon as possible the non-optimality of a first-stage candidate, it solves only a small proportion of the subproblems at most iterations. We also propose a general framework to stabilize our algorithm, and show its finite convergence and exact behavior. We report an extensive computational study on large-scale instances of stochastic optimization literature that shows the efficiency of the proposed algorithm compared to six alternative algorithms from the literature (monocut and multicut implementation of the Benders decomposition algorithm with and without an in-out stabilization approach, a monocut implementation of a level bundle method, IBM ILOG CPLEX 12.10 built-in Benders decomposition). We also obtain significant additional computational time savings using the primal stabilization schemes.

We have studied a class of two-stage robust binary optimization problems with objective uncertainty where recourse decisions are restricted to be mixed-binary 1. For these problems, we present a deterministic equivalent formulation through the convexification of the recourse feasible region. We then explore this formulation under the lens of a relaxation, showing that the specific relaxation we propose can be solved using the branch-and-price algorithm. We present conditions under which this relaxation is exact, and describe alternative exact solution methods when this is not the case. Despite the two-stage nature of the problem, we provide NP-completeness results based on our reformulations. Finally, we present various applications in which the methodology we propose can be applied. We compare our exact methodology to those approximate methods recently proposed in the literature under the name K-adaptability. Our computational results show that our methodology is able to produce better solutions in less computational time compared to the K-adaptability approach, as well as to solve bigger instances than those previously managed in the literature.

We further extend this work in 22, where we address general problems in which all constraints (including those linking the first and the second stages) are defined by convex functions and involve mixed-integer variables, thus extending the existing literature to a much wider class of problems. We show how these problems can be reformulated using Fenchel duality, allowing to derive an enumerative exact algorithm, for which we prove $ϵ-$convergence in a finite number of operations. An implementation of the resulting algorithm, embedding a column generation scheme, is then computationally evaluated on two different problems, using instances that are derived starting from the existing literature. To the best of our knowledge, this is the first approach providing results on the practical solution of this class of problems.

8.2 Arc-flow models

Network flow formulations are among the most successful tools to solve optimization problems. Such formulations correspond to determining an optimal flow in a network. One particular class of network flow formulations is the arc flow, where variables represent flows on individual arcs of the network. For hard problems, polynomial-sized arc-flow models typically provide weak linear relaxations and may have too much symmetry to be efficient in practice. Instead, arc flow models with a pseudo-polynomial size usually provide strong relaxations and are efficient in practice. The interest in pseudo-polynomial arc-flow formulations has grown considerably in the last twenty years, in which they have been used to solve many open instances of hard problems. A remarkable advantage of pseudo-polynomial arc-flow models is the possibility to solve practical-sized instances directly by a Mixed Integer Linear Programming solver, avoiding the implementation of complex methods based on column generation.

In 8, we present theoretical foundations of pseudo-polynomial arc-flow formulations, by showing a relation between their network and Dynamic Programming (DP). This relation allows a better understanding of the strength of these formulations, through a link with models obtained by Dantzig-Wolfe decomposition. The relation with DP also allows a new perspective to relate state-space relaxation methods for DP with arc-flow models. We also present a dual point of view to contrast the linear relaxation of arc-flow models with that of models based on paths and cycles. To conclude, we review the main solution methods and applications of arc-flow models based on DP in several domains such as cutting, packing, scheduling, and routing.

8.3 Machine scheduling problems

In 23, we study the prize-collecting job sequencing problem with one common and multiple secondary resources. In this problem, a set of jobs is given, each with a profit, multiple time windows for its execution, and a duration during which it requires the main resource. Each job also requires one of the secondary resources before, during, and after its use of the main resource. The goal is to select and schedule the subset of jobs that maximize the total profit. The problem has application in particle therapy scheduling and in pre-runtime scheduling of avionic systems among others. We present a new mixed integer linear programming formulation of the problem and a branch-cut-and-price algorithm as exact solution methods. We also introduce a heuristic algorithm to tackle larger instances. Extensive numerical experiments show that our exact algorithms can solve to optimality literature instances with up to 500 jobs for a particular dataset and up to 250 jobs for another dataset with different characteristics. Our heuristic builds high-quality solutions in a small computational time. It computes new best-known solutions for most of the larger instances.

8.4 Vehicle routing problems

The Shortest Path Problem with Resource Constraints (SPPRC) arises as a subproblem in state-of-the-art Branch-Cut-and-Price algorithms for vehicle routing problems, including the BCP solver described just above. In 5, we propose a variant of the bi-directional label correcting algorithm in which the labels are stored and extended according to the so-called bucket graph. Such organization of labels helps to decrease significantly the number of dominance checks and the running time of the algorithm. We also show how the forward/backward route symmetry can be exploited and how to eliminate arcs from the bucket graph using reduced costs. The proposed algorithm can be especially beneficial for vehicle routing instances with large vehicle capacity and/or with time window constraints. Computational experiments were performed on instances from the distance constrained vehicle routing problem, including multi-depot and site-dependent variants, on the vehicle routing problem with time windows, and on the "nightmare" instances of the heterogeneous fleet vehicle routing problem. Significant improvements over the best algorithms in the literature were achieved and many instances could be solved for the first time. The proposed algorithm is the central part of the generic solver for vehicle routing and related problems proposed in 4. This solver is based on BaPCod library 24 which is being developed in the team.

In 3, we examine the robust counterpart of the classical Capacitated Vehicle Routing Problem (CVRP). We consider two types of uncertainty sets for the customer demands: the classical budget polytope introduced by Bertsimas and Sim (2003), and a partitioned budget polytope proposed by Gounaris et al. (2013). We show that using the set-partitioning formulation it is possible to reformulate our problem as a deterministic heterogeneous vehicle routing problem. Thus, many state-of-the-art techniques for exactly solving deterministic VRPs can be applied for the robust counterpart, and a modern branch-and-cut-and-price algorithm can be adapted to our setting by keeping the number of pricing subproblems strictly polynomial. More importantly, we introduce new techniques to significantly improve the efficiency of the algorithm. We present analytical conditions under which a pricing subproblem is infeasible. This result is general and can be applied to other combinatorial optimization problems with knapsack uncertainty. We also introduce robust capacity cuts which are provably stronger than the ones known in the literature. Finally, a fast iterated local search algorithm is proposed to obtain heuristic solutions for the problem. Using our branch-and-cut-and-price algorithm incorporating existing and new techniques, we are able to solve to optimality all but one open instances from the literature.

In 14, we propose a partial optimization metaheuristic under special intensification conditions (POPMUSIC) for the classical capacitated vehicle routing problem (CVRP). The proposed approach uses a branch-cut-and-price algorithm as a powerful heuristic to solve subproblems whose dimensions are typically between 25 and 200 customers. The whole algorithm can be seen as the application of local search over very large neighborhoods, starting from a single initial solution. The main computational experiments were carried out on instances having between 302 and 1000 customers. Using initial solutions generated by some of the best available metaheuristics for the problem, POPMUSIC was able to obtain consistently better solutions for long runs of up to 32 hours. In a final experiment, starting from the best known solutions available in CVRP library (CVRPLIB), POPMUSIC was able to find new best solutions for several instances, including some very large ones.

8.5 Cutting and packing problems

In 15, we introduce and motivate a variant of the bin packing problem where bins are assigned to time slots, and minimum and maximum lags are required between some pairs of items. We suggest two integer programming formulations for the problem: a compact one, and a stronger formulation with an exponential number of variables and constraints. We propose a branch-cut-and-price approach which exploits the latter formulation. For this purpose, we devise separation algorithms based on a mathematical characterization of feasible assignments for two important special cases of the problem. Computational experiments are reported for instances inspired from a real-case application of chemical treatment planning in vineyards, as well as for literature instances for special cases of the problem. The experimental results show the efficiency of our branch-cut-and-price approach, as it outperforms the compact formulation of newly proposed instances, and is able to obtain improved lower and upper bounds for literature instances.

In 13, we propose branch-cut-and-price algorithms for the classic bin packing problem and also for the following related problems: vector packing, variable sized bin packing and variable sized bin packing with optional items. The algorithms are defined as models for VRPSolver, a generic solver for vehicle routing problems. In that way, a simple parameterization enables the use of several branch-cut-and-price advanced elements: automatic stabilization by smoothing, limited-memory rank-1 cuts, enumeration, hierarchical strong branching and limited discrepancy search diving heuristics. As an original theoretical contribution, we prove that the branching over accumulated resource consumption, that does not increase the difficulty of the pricing subproblem, is sufficient for those bin packing models. Extensive computational results on instances from the literature show that the VRPSolver models have a performance that is very robust over all those problems, being often superior to the existing exact algorithms on the hardest instances. Several instances could be solved to optimality for the first time.

We have developed an approach to solve the temporal knapsack problem (TKP) based on a very large size dynamic programming formulation 7. In this generalization of the classical knapsack problem, selected items enter and leave the knapsack at fixed dates. We solve the TKP with a dynamic program of exponential size, which is solved using a method called Successive Sublimation Dynamic Programming (SSDP). This method starts by relaxing a set of constraints from the initial problem, and iteratively reintroduces them when needed. We show that a direct application of SSDP to the temporal knapsack problem does not lead to an effective method, and that several improvements are needed to compete with the best results from the literature.

8.6 Network Design Problems

More than ever, data networks have demonstrated their central role in the world economy, but also in the well-being of humanity that needs fast and reliable networks. In parallel, with the emergence of Network Function Virtualization (NFV) and Software Defined Networking (SDN), efficient network algorithms considered too hard to be put in practice in the past now have a second chance to be considered again. In this context, as new networks will be deployed and current ones get significant upgrades, it is thus time to rethink the network dimensioning problem with protection against failures. In 17, we consider a path-based protection scheme with the global rerouting strategy in which, for each failure situation, there may be a new routing of all the demands. Our optimization task is to minimize the needed amount of bandwidth. After discussing the hardness of the problem, we develop two scalable mathematical models that we handle using both Column Generation and Benders Decomposition techniques. Through extensive simulations on real-world IP network topologies and on randomly generated instances, we show the effectiveness of our methods: they lead to savings of 40 to 48% of the bandwidth to be installed in a network to protect against failures compared to traditional schemes. Finally, our implementation in OpenDaylight demonstrates the feasibility of the approach. Its evaluation with Mininet shows that our solution provides sub-second recovery times, but the way it is implemented may greatly impact the amount of signaling traffic exchanged. In our evaluations, the recovery phase requires only a few tens of milliseconds for the fastest implementation, compared to a few hundreds of milliseconds for the slowest one.

8.7 Energy

Optimizing nuclear unit outages is of significant economic importance for the French electricity company EDF, as these outages induce a substitute production by other more expensive means to fulfill electricity demand. This problem is quite challenging given the specific operating constraints of nuclear units, the stochasticity of both the demand and non-nuclear units availability, and the scale of the instances. To tackle these difficulties we use a combined decomposition approach in 10. The operating constraints of the nuclear units are built into a Dantzig-Wolfe pricing subproblem whose solutions define the columns of a demand covering formulation. The scenarios of demand and non-nuclear units availability are handled in a Benders decomposition. Our approach is shown to scale up to the real-life instances of the French nuclear fleet.

8.8 Sustainable agriculture

In 20, we investigate the robust planning and scheduling of activities in agriculture and in particular the application of phytosanitary treatments. The crops are subject to many diseases that may arise during different time windows of the planning horizon. In response, a phytosanitary treatment can be applied to protect against a subset of these diseases. However, the effective duration of some treatments is uncertain, it depends on the type of treatment applied as well as on the weather conditions. In this study we introduce a penalty function based approach to handle this uncertainty without being overly conservative akin to light robustness approach proposed in the literature. We discuss different forms for this penalty function and elaborate on solution methodologies for the resulting models. We test the effectiveness of our approach with realistically-sized instances, which correspond to a typical vineyard in Bordeaux area, and present a numerical analysis of different optimization models and solution methods.

Our work in 15 cited above also finds its practical application in the context of application of phytosanitary treatments.

9.1 Bilateral contracts with industry

We have a contract with RTE to develop strategies inspired from stochastic gradient methods to speed-up Benders' decomposition. The PhD thesis of Xavier Blanchot is part of this contract.

9.2 Bilateral grants with industry

Our joint project with Atoptima start-up "Solution methods for the inventory routing problem: application to waste collection in the urban environment" has been supported in 2020 by Nouvelle Aquitaine region (appel à projet "Recherche et Enseignement Supérieur"). The project is financing one half of a PhD thesis.

We received the funding for a one year post-doctoral position from the PGMO fundation, for a joint project with EDF. This project aims at improving robustness and stability of power plants planning given the uncertainty associated with production fleet availability (nuclear outage durations, nuclear availability, growth of renewable sources, imponderable events such as the Covid-19 crisis...).

10.1 National initiatives

10.2 Regional initiatives

11.1 Promoting scientific activities

11.2 Teaching - Supervision - Juries

11.3 Popularization

12.1 Major publications

• 1 articleA. N.Ayşe N Arslan and B.Boris Detienne. Decomposition-based approaches for a class of two-stage robust binary optimization problems.INFORMS Journal on Computing2021
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• 2 articleF.François Clautiaux, S.Said Hanafi, R.Rita Macedo, M.-E.Marie-Emilie Voge and C.Cláudio Alves. Iterative aggregation and disaggregation algorithm for pseudo-polynomial network flow models with side constraints.European Journal of Operational Research2582017, 467 - 477
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• 3 articleA. A.Artur Alves Pessoa, M.Michael Poss, F.François Vanderbeck, R.Ruslan Sadykov and F.Francois Vanderbeck. Branch-and-cut-and-price for the robust capacitated vehicle routing problem with knapsack uncertainty.Operations Research6932021, 739-754
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• 4 articleA. A.Artur Alves Pessoa, R.Ruslan Sadykov, E.Eduardo Uchoa and F.François Vanderbeck. A Generic Exact Solver for Vehicle Routing and Related Problems.Mathematical Programming1832020, 483-523
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• 5 articleR.Ruslan Sadykov, A. A.Artur Alves Pessoa and E.Eduardo Uchoa. A Bucket Graph Based Labelling Algorithm for Vehicle Routing.Transportation Science5512021, 4-28
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12.2 Publications of the year

12.3 Cited publications

• 25 unpublishedA. J.Andrew J. Akartunali. A Computational Analysis of Lower Bounds for Big Bucket Production Planning Problems.2009, URL: http://hal.archives-ouvertes.fr/hal-00387105/en/
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• 26 articleA. J.Andrew J. Akartunali. A heuristic approach for big bucket multi-level production planning problems.European Journal of Operational Research2009, 396--411URL: http://hal.archives-ouvertes.fr/hal-00387052/en/
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• 27 articleM.Manuel Alba Martínez, F.François Clautiaux, M.Mauro Dell'Amico and M.Manuel Iori. Exact algorithms for the bin packing problem with fragile objects.Discrete Optimization103August 2013, 210--223URL: http://hal.inria.fr/hal-00909480
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• 28 articleR.Ruslan Baptiste. On Scheduling a Single Machine to Minimize a Piecewise Linear Objective Function : A Compact MIP Formulation.Naval Research Logistics / Naval Research Logistics An International Journal5662009, 487--502URL: http://hal.inria.fr/inria-00387012/en/
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• 29 articleR.Ruslan Baptiste. Time Indexed Formulations for Scheduling Chains on a Single Machine: An Application to Airborne Radars.European Journal of Operational Research2009, URL: http://hal.inria.fr/inria-00339639/en/
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• 30 techreportF.François Clautiaux, R.Ruslan Sadykov, F.François Vanderbeck and Q.Quentin Viaud. Pattern based diving heuristics for a two-dimensional guillotine cutting-stock problem with leftovers.Université de BordeauxDecember 2017, 1--30
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• 31 articleA. J.Andrew J. AND Van Vyve Constantino. Mixing MIR Inequalities with Two Divisible Coefficients.Mathematical Programming, Series A2009, 1--1URL: http://hal.archives-ouvertes.fr/hal-00387098/en/
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• 32 articleA. J.Andrew J. AND Balasubramanian Denton. Optimal Allocation of Surgery Blocks to Operating Rooms Under Uncertainty.Operations Research2009, 1--1URL: http://hal.archives-ouvertes.fr/hal-00386469/en/
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• 33 inproceedingsB.Boris Detienne. Extended formulations for robust maintenance planning at power plants.Gaspard Monge Program for Optimization : Conference on Optimization and Practices in Industry PGMO-COPI14Saclay, FranceOctober 2014
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• 34 inproceedingsM. T.M. T. Godinho, L.L. Gouveia, T. L.T. L. Magnanti, P.Pierre Pesneau and J.J. Pires. On Time-Dependent Model for Unit Demand Vehicle Routing Problems.International Conference on Network Optimization, INOCSpa, BelgiumInternational Network Optimization Conference (INOC)2007
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• 35 techreportM. T.M. T. Godinho, L.L. Gouveia, T. L.T. L. Magnanti, P.Pierre Pesneau and J.J. Pires. On a Time-Dependent Model for the Unit Demand Vehicle Routing Problem.11-2007Centro de Investigacao Operacional da Universidade de Lisboa2007
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• 36 inproceedingsR.Rodolphe Griset, P.Pascale Bendotti, B.Boris Detienne, H.Hugo Grevet, M.Marc Porcheron, H.Halil Şen and F.François Vanderbeck. Efficient formulations for nuclear outages using price and cut, Snowcap project..PGMO Days 2017Saclay, FranceNovember 2017
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• 37 inproceedingsR.Rodolphe Griset, P.Pascale Bendotti, B.Boris Detienne, H.Hugo Grevet, M.Marc Porcheron and F.François Vanderbeck. Scheduling nuclear outage with cut and price (Snowcap).Mathematical Optimization in the Decision Support Systems for Efficient and Robust Energy Networks Final ConferenceModena, ItalyMarch 2017
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• 38 inproceedingsR.Rodolphe Griset. Optimisation des arrêts nucléaires : une amélioration des solutions développées par EDF suite au challenge ROADEF 2010.18ème conférence de la société française de recherche opérationnelle et d'aide à la décision ROADEF 2017Metz, FranceFebruary 2017
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• 39 articleY.Y. Guan, S.S. Ahmed, A. J.Andrew J. Miller and G.G.L. Nemhauser. On formulations of the stochastic uncapacitated lot-sizing problem.Operations Research Letters342006, 241--250
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• 40 articleY.Y. Guan, S.S. Ahmed, G.G.L. Nemhauser and A. J.Andrew J. Miller. A branch-and-cut algorithm for the stochastic uncapacitated lot-sizing problem.Mathematical Programming1052006, 55--84
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• 41 inproceedingsJ.Jinil Han, P.Pascale Bendotti, B.Boris Detienne, G.Georgios Petrou, M.Marc Porcheron, R.Ruslan Sadykov and F.François Vanderbeck. Extended Formulation for Maintenance Planning at Power Plants.ROADEF - 15ème congrès annuel de la Société française de recherche opérationnelle et d'aide à la décisionSociété française de recherche opérationnelle et d'aide à la décisionBordeaux, FranceFebruary 2014
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• 42 inproceedingsY.Y. Hendel and R.Ruslan Sadykov. Timing problem for scheduling an airborne radar.Proceedings of the 11th International Workshop on Project Management and SchedulingIstanbul, TurkeyApril 2008, 132--135
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• 43 articleD.D. Huygens, M.Martine Labbé, A. R.A. R. Mahjoub and P.Pierre Pesneau. The two-edge connected hop-constrained network design problem: Valid inequalities and branch-and-cut.Networks4912007, 116--133
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• 44 articleM.~Mourgaya and F.François Vanderbeck. Column generation based heuristic for tactical planning in multi period vehicle routing.European Journal of Operational Research18332007, 1028--1041
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• 45 articleM.~Padberg and G.~Rinaldi. A branch-and-cut algorithm for the resolution of large-scale symmetric traveling salesman problems.SIAM Review3311991, 60--100
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• 46 inproceedingsP. A.Pierre AND Vanderbeck Meurdesoif. A Branch­-and­-Cut algorithm to optimize sensor installation in a network.Graph and Optimization Meeting GOM2008France Saint-Maximin2008
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• 47 articleN. A.Nancy AND Vanderbeck Michel. Knapsack Problems with Setups.European Journal of Operational Research1962009, 909--918URL: http://hal.inria.fr/inria-00232782/en/
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• 48 articleS.Sophie Michel and F.François Vanderbeck. A Column Generation based Tactical Planning Method for Inventory Routing.Operations Research6022012, 382--397
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• 49 inproceedingsP.Pierre Pesneau, F.François Clautiaux and J.Jeremy Guillot. Aggregation technique applied to a clustering problem.4th International Symposium on Combinatorial Optimization (ISCO 2016)Vietri sul Mare, ItalyMay 2016
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• 50 inproceedingsP.Pierre Pesneau, F.François Clautiaux and J.Jeremy Guillot. Aggregation technique applied to a clustering problem for waste collection..ROADEF 2016Compiègne, FranceFebruary 2016
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• 51 inproceedingsM.Marcus Poggi, D.Diego Pecin, M.M. Reis, C.C. Ferreira, K.K. Neves, R.Ruslan Sadykov and F.François Vanderbeck. Equipment/Operator task scheduling with BAPCOD.Column Generation 2012Bromont, CanadaJune 2012
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• 52 inproceedingsN.Nastaran Rahmani, B.Boris Detienne, R.Ruslan Sadykov and F.François Vanderbeck. A Column Generation Based Heuristic for the Dial-A-Ride Problem.International Conference on Information Systems, Logistics and Supply Chain (ILS)Bordeaux, FranceJune 2016
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• 53 articleR.Ruslan Sadykov. A branch-and-check algorithm for minimizing the sum of the weights of the late jobs on a single machine with release dates.European Journal of Operations Research18932008, 1284--1304URL: http://dx.doi.org/10.1016/j.ejor.2006.06.078
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• 54 techreportR.Ruslan Sadykov. A polynomial algorithm for a simple scheduling problem at cross docking terminals.RR-7054INRIA2009, URL: http://hal.inria.fr/inria-00412519/en/
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• 55 inproceedingsR.Ruslan Sadykov. On scheduling malleable jobs to minimise the total weighted completion time.13th IFAC Symposium on Information Control Problems in ManufacturingRussie Moscow2009, URL: http://hal.inria.fr/inria-00339646/en/
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• 56 articleR.Ruslan Sadykov. Scheduling incoming and outgoing trucks at cross docking terminals to minimize the storage cost.Annals of Operations Research20112012, 423--440
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• 57 articleR.Ruslan Sadykov and F.François Vanderbeck. Bin Packing with conflicts: a generic branch-and-price algorithm.INFORMS Journal on Computing2522013, 244--255URL: http://hal.inria.fr/inria-00539869
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• 58 inproceedingsL. G.L. Gély AND G. Dessagne AND Pierre Pesneau AND François Vanderbeck. A multi scalable model based on a connexity graph representation.11th International Conference on Computer Design and Operation in the Railway and Other Transit Systems COMPRAIL'08Toledo, SpainSeptember 2008
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• 59 phdthesisB.Benoit Vignac. Résolution d'un problème de groupage dans le réseaux optiques maillés.Université de MontréalJanuary 2010
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• 60 inproceedingsL.L. ~Gély. Real-time train scheduling at SNCF.1st Workshop on Robust Planning and Rescheduling in RailwaysUtrechtARRIVAL meeting on Robust planning and Rescheduling in RailwaysApril 2007
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