## Section: New Results

### Large scale complex structure optimization

**Joint order batching and picker routing problem:** Order picking is the process of retrieving products from inventory.
It is mostly done manually by dedicated employees called pickers and is considered the most expensive of warehouse operations.
To reduce the picking cost, customer orders can be grouped into batches that are then collected by traveling the shortest possible distance. We proposed an industrial case study for the HappyChic company where the warehouse has an acyclic layout: pickers are not allowed to backtrack.
We developed a two-phase heuristic approach to solve this industrial case [64].
Moreover, we propose an exponential linear programming formulation to tackle the joint order batching and picker routing problem.
Variables, or columns, are related to the picking routes in the warehouse.
Computing such routes is generally an intractable routing problem and relates to the well known traveling salesman problem (TSP).
Nonetheless, the rectangular warehouse's layouts can be used to efficiently solve the corresponding TSP and take into account in the development of an efficient subroutine, called oracle.
We therefore investigate whether such an oracle allows for an effective exponential formulation.
Experimented on a publicly available benchmark, the algorithm proves to be very effective.
It improves many of the best known solutions and provides very strong lower bounds.
Finally, this approach is also applied to the HappyChic industrial case to demonstrate its interest for this field of application [67], [45].

**Logistics network design problem:**
Planning transportation operations within a supply chain is a difficult task that is often outsourced to logistics providers, in practice. At the tactical level, the problem of distributing products through a multi-echelon network is defined in the literature as the Logistics Service Network Design Problem (LSNDP). We study a LSNDP variant inspired by the management of restaurant supply chains. In this problem, a third party carrier seeks to cost-effectively source and fulfill customer demands of products through a tri-echelon supply chain composed of suppliers, warehouses, and customers. We propose an exact solution method based on partial Benders decompositions, where the master problem is strengthened by the addition of aggregated information derived from the subproblem. More specifically, we introduce a high-level dynamic Benders approach where the aggregated information used to strengthen the master is refined iteratively. In an extensive computational study, we demonstrate that our dynamic Benders strategy produces provably high-quality solutions and we validate the interest of refining the master problem in the course of a partial Benders decomposition-based scheme [72].

**Multi commodity vehicle routing problem:**
We study vehicle routing problems considering multiple commodities, with applications in the local fresh food supply chains.
The studied supply chain contains two echelons with three sets of actors: suppliers, distribution centers and customers.
Suppliers are farmers that produce some fresh foods.
Distribution centers are in charge of consolidation and delivery of the products to customers.
Distribution centers collect products from the suppliers that perform direct trips.
Products are delivered to the customers with a fleet of vehicles performing routes.
Each customer requires several commodities, and the farmers produce a limited quantity of these commodities.
For the minimization of the transportation cost, it is beneficial that a single customer is delivered by several vehicles.
However, for the convenience of the customer, it is imposed that a single commodity is delivered at once by a single vehicle.
Hence, different commodities have been explicitly considered.
The complete problem is named Multi-Commodity two-echelon Distribution Problem (MC2DP).
The restricted problem that addresses only the delivery from a single distribution center is named Commodity constrained Split Delivery Vehicle Routing Problem (C-SDVRP).
We first propose a heuristic based on the Adaptive Large Neighborhood Search (ALNS) for the C-SDVRP [22].
Then, we address the whole problem (MC2DP) with collection and delivery operations and multiple distribution centers.
In order to tackle this complex problem, we propose to decompose the problem: collection and delivery are sequentially solved [53], [54].

**Generalized routing problems:** We study routing problems that arise in the context of last mile delivery when multiple delivery options are proposed to the customers.
The most common option to deliver packages is home/workplace delivery. Besides, the delivery can be made to pick-up points such as dedicated lockers or stores. In recent years, a new concept called trunk/in-car delivery has been proposed.
Here, customers' packages can be delivered to the trunks of cars.
Our goal is to model and develop efficient solution approaches for routing problems in this context, in which each customer can have multiple shipping locations. First, we study the single-vehicle case in the considered context, which is modeled as a Generalized Traveling Salesman Problem with Time Windows (GTSPTW).
Four mixed integer linear programming formulations and an efficient branch-and-cut algorithm are proposed.
Then, we study the multi-vehicle case which is denoted Generalized Vehicle Routing Problem with Time Windows (GVRPTW).
An efficient column generation based heuristic is proposed to solve it [60], [61], [59], [37].

**Joint management of demand and offer for last-mile delivery systems:**
E-commerce is a thriving market around the world and suits very well the busy lifestyle of today's customers. This growing in e-commerce poses a huge challenge for transportation companies, especially in the last mile delivery. We addressed first a fleet composition problem for last-mile delivery service. This problem occurs at a tactical level when the composition of the fleet has to be decided in advance. It is the case for companies that offer last-mile delivery service. Most of them subcontract the transportation part to local carriers and have to decide the day before which vehicles will be needed to cover a partially known demand. We assumed that the distribution area is divided into a limited number of delivery zones and the time horizon into time-slots. The demand is characterized by packages to be transported from pick-up zones to delivery zones given a delivery time slot. We have proposed an optimization problem aiming jointly to manage the offer and the demand. More precisely, discrete choice models representing the choices made by couriers and customers are integrated into the same optimization model.
The originality of the contribution is based on the integration of variables of different natures in the same model and the development of integrated resolution methods. On the basis of a closed form of discrete choice models, we have reformulated the problem as a non-linear optimization problem. The resolution of this model by classical solvers requires coupling exact methods with heuristics in order to define a first initial solution [47], [58].

**Delay Management in Public Transportation:**
The Delay Management Problem arises in Public Transportation networks, and is characterized by the necessity of connections between different vehicles. The attractiveness of Public Transportation networks is strongly related to the reliability of connections, which can be missed when delays or other unpredictable events occur. Given a single initial delay at one node of the network, the Delay Management Problem is to determine which vehicles have to wait for the delayed ones, with the aim of minimizing the dissatisfaction of the passengers. We derived strengthened mixed integer linear programming formulations and new families of valid inequalities for that problem. The implementation of branch-and-cut methods and tests on a benchmark of instances taken from real networks show the potential of the proposed formulations and cuts [20].

**Discrete Ordered Median Problem:**
The discrete ordered median problem consists in locating $p$ facilities in order to minimize an ordered weighted sum of distances between clients and closest open facility. We formulate this problem as a set partitioning problem using an exponential number of variables. Each variable corresponds to a set of demand points allocated to the same facility with the information of the sorting position of their corresponding costs. We develop a column generation approach to solve the continuous relaxation of this model. Then, we apply a branch-price-and-cut algorithm to solve small to large sized instances of DOMP in competitive computational time [21].

**Genome wide association studies:**
We studied the Polymorphic Alu
Insertion Recognition Problem (PAIRP). Alu (Arthrobacter luteus) forms a major component of repetitive DNA and are frequently encountered during the genotyping of individuals. The basic approach to find Alus consists of (1) aligning sequence reads from a set of individual(s) with respect to a reference genome and (2) comparing the possible Alu insertion induced by the alignment with the Alu insertions positions already known for the reference genome. The sequence genome of the reference individual is known and will be highly
similar, but not identical, to the genome of the individual(s) being sequenced.
Hence, at some locations they will diverge. Some of this divergence is due to the insertion of Alu polymorphisms. Detecting Alus has a central role in the field of Genetic Wide Association Studies because basic elements are a common source of mutation in humans. We investigated the PAIRP relationship with the the Clique Partitioning of Interval Graphs (CPIG). Our results [29] provide insights of the complexity of the problem, a characterization of its combinatorial structure and an exact approach based on Integer Linear Programming to exactly solve the correspond instances.

**A branch-and-cut algorithm for the maximum k-balanced subgraph of a signed graph:**
A signed graph is $k$-balanced if its vertex set can be partitioned into at most $k$ sets in such a way that positive edges are found only within the sets and negative edges go between sets. The maximum $k$-balanced subgraph problem is the problem of finding a subgraph of a graph $G$ that is $k$-balanced and maximum according to the number of vertices. This problem has applications in clustering problems appearing in collaborative vs conflicting environments. We provide a representatives formulation for the problem and present a partial description of the associated polytope, including the introduction of strengthening families of valid inequalities. A branch-and-cut algorithm is described for finding an optimal solution to the problem. An ILS metaheuristic is implemented for providing primal bounds for this exact method and a branching rule strategy is proposed for the representatives formulation. Computational experiments, carried out over a set of random instances and on a set of instances from an application, show the effectiveness of the valid inequalities and strategies adopted in this work [75].

**Feature Selection in Support Vector Machine:**
This work focuses on support vector machine (SVM) with feature selection. A MILP formulation is proposed for the problem. The choice of suitable features to construct the separating hyperplanes has been modelled in this formulation by including a budget constraint that sets in advance a limit on the number of features to be used in the classification process. We propose both an exact and a heuristic procedure to solve this formulation in an efficient way. Finally, the validation of the model is done by checking it with some well-known data sets and comparing it with classical classification methods [24].