Cécile Murat

Robust location transportation problems under uncertain demands

In robust optimization, the multi-stage context (or dynamic decision-making) assumes that the information is revealed in stages. So, part of the decisions must be taken before knowing the real values of the uncertain parameters, and another part, called recourse decisions, is taken when the information is known. In this paper, we are interested in a robust version of the location transportation problem with an uncertain demand using a 2-stage formulation. The obtained robust formulation is a convex (not linear) program, and we apply a cutting plane algorithm to exactly solve the problem. At each iteration, we have to solve an NP-hard recourse problem in an exact way, which is time-consuming. Here, we go further in the analysis of the recourse problem of the location transportation problem. In particular, we propose a mixed integer program formulation to solve the quadratic recourse problem and we define a tight bound for this reformulation. We present an extensive computation analysis of the 2-stage robust location transportation problem. The proposed tight bound allows us to solve large size instances.

A new single model and derived algorithms for the satellite shot planning problem using graph theory concepts

The satellite shot sequencing problem consists in choosing the pictures to be completed by defining sequences of shots which must respect technical constraints and limits. We propose a graph-theoretic model for both the medium- and the short-term sequencing and present algorithmic solutions by using properties of the model.

The probabilistic longest path problem

We study the probabilistic longest path problem. We propose a modification strategy adapting a solution for a deterministic instance to a solution for the probabilistic one, we compute the functional associated with this strategy, and we evaluate the complexities of computing this functional and of computing the deterministic solution maximizing it.

Linear programming with interval right hand sides

In this paper, we study general linear programs in which right handsides are interval numbers. This model is relevant when uncertain and inaccurate factors make di±cult the assignment of a single value to each right handside. When objective function coefficients are interval numbers in a linear program, it is used to determine optimal solutions according to classical criteria coming from decision theory (like the worst case criterion). When the feasible solutions set is uncer- tain, another approach consists in determining the worst and best optimum solutions. We study the complexity of these two optimization problems when each right handside is an interval number. Moreover, we analysis the relationship between these two problems and the classical approach coming from decision theory. We exhibit some duality relation between the worst optimum solution problem and the best optimum solution problem in the dual. This study highlights some duality property in robustness analysis.

Mathematical Programming for Earth Observation Satellite Mission Planning

Planning the mission of an Earth observation satellite is choosing the shots to be taken during a given period in order to satisfy some requested images. The difficulty of the underlying combinatorial problem depends on the satellite characteristics and on the planning horizon.

A priori optimization for the probabilistic maximum independent set problem

We first propose a formal definition for the concept of probabilistic combinatorial optimization problem (under the a priori method). Next, we study the complexity of optimally solving probabilistic maximum independent set problem under several a priori optimization strategies as well as the complexity of approximating optimal solutions. For the different strategies studied, we present results about the restriction of probabilistic independent set on bipartite graphs.

Robustness and duality in linear programming

AbstractIn this paper, we consider a linear program in which the right hand sides of the constraints are uncertain and inaccurate. This uncertainty is represented by intervals, that is to say that each right hand side can take any value in its interval regardless of other constraints. The problem is then to determine a robust solution, which is satisfactory for all possible coefficient values. Classical criteria, such as the worst case and the maximum regret, are applied to define different robust versions of the initial linear program. More recently, Bertsimas and Sim have proposed a new model that generalizes the worst case criterion. The subject of this paper is to establish the relationships between linear programs with uncertain right hand sides and linear programs with uncertain objective function coefficients using the classical duality theory. We show that the transfer of the uncertainty from the right hand sides to the objective function coefficients is possible by establishing new duality relations. When the right hand sides are approximated by intervals, we also propose an extension of the Bertsimas and Sims model and we show that the maximum regret criterion is equivalent to the worst case criterion.

The probabilistic minimum vertex covering problem

An instance of the probabilistic vertex-covering problem is a pair (G=(V,E),Pr) obtained by associating with each vertex υ[sub i] ∈V an ‘occurrence’ probability p[sub i] . We consider a modification strategy Μ transforming a vertex cover C for G into a vertex cover C[sub I] for the subgraph of G induced by a vertex-set I⊆V. The objective for the probabilistic vertex-covering is to determine a vertex cover of G minimizing the sum, over all subsets I⊆V, of the products: probability of I times C[sub I] . In this paper, we study the complexity of optimally solving probabilistic vertex-covering.

On the probabilistic minimum coloring and minimum k -coloring

We study a robustness model for the minimum coloring problem, where any vertex vi of the input-graph G(V, E) has some presence probability Pi. We show that, under this model, the original coloring problem gives rise to a new coloring version (called Probabilistic Min Coloring) where the objective becomes to determine a partition of V into independent sets S1, S2,..., Sk, that minimizes the quantity Σi=1k f(Si), where, for any independent set Si, i = 1,..., k, f(Si) = 1 - Πvj∈si (1 - pj). We show that Probabilistic Min Coloring is NP-hard and design a polynomial time approximation algorithm achieving non-trivial approximation ratio. We then focus ourselves on probabilistic coloring of bipartite graphs and show that the problem of determining the best k-coloring (called Probabilistic Min k-Coloring) is NP-hard, for any k ≥ 3. We finally study Probabilistic Min Coloring and Probabilistic Min k-Coloring in a particular family of bipartite graphs that plays a crucial role in the proof of the NP-hardness result just mentioned, and in complements of bipartite graphs.

New models for the robust shortest path problem: complexity, resolution and generalization

In optimization, it is common to deal with uncertain and inaccurate factors which make it difficult to assign a single value to each parameter in the model. It may be more suitable to assign a set of values to each uncertain parameter. A scenario is defined as a realization of the uncertain parameters. In this context, a robust solution has to be as good as possible on a majority of scenarios and never be too bad. Such characterization admits numerous possible interpretations and therefore gives rise to various approaches of robustness. These approaches differ from each other depending on models used to represent uncertain factors, on methodology used to measure robustness, and finally on analysis and design of solution methods. In this paper, we focus on the application of a recent criterion for the shortest path problem with uncertain arc lengths. We first present two usual uncertainty models: the interval model and the discrete scenario set model. For each model, we then apply a criterion, called bw-robustness (originally proposed by B. Roy) which defines a new measure of robustness. According to each uncertainty model, we propose a formulation in terms of large scale integer linear program. Furthermore, we analyze the theoretical complexity of the resulting problems. Our computational experiments perform on a set of large scale graphs. By observing the results, we can conclude that the approved solvers, e.g. Cplex, are able to solve the mathematical models proposed which are promising for robustness analysis. In the end, we show that our formulations can be applied to the general linear program in which the objective function includes uncertain coefficients.