Céline Gicquel

A second-order cone programming approximation to joint chance-constrained linear programs

We study stochastic linear programs with joint chance constraints, where the random matrix is a special triangular matrix and the random data are assumed to be normally distributed. The problem can be approximated by another stochastic program, whose optimal value is an upper bound of the original problem. The latter stochastic program can be approximated by two second-order cone programming (SOCP) problems [5]. Furthermore, in some cases, the optimal values of the two SOCPs problems provide a lower bound and an upper bound of the approximated stochastic program respectively. Finally, numerical examples with probabilistic lot-sizing problems are given to illustrate the effectiveness of the two approximations.

A distribution network design problem in the automotive industry

We consider a multi-product distribution network design problem arising from a case-study in the automotive industry. Based on the realistic assumptions, we introduce minimum volume, maximum covering distance and single sourcing constraints, making the problem difficult to solve for large-size instances. We thus develop several heuristic procedures using various relaxations of the original MIP formulation of the problem. In our numerical experiments, we analyze the structure of the obtained network as well as the impact of varying the problem parameters on computation times. We also show that the implemented heuristic methods provide good quality solutions within short computation times on instances for which a state-of-the-art MIP solver does not produce any feasible solution.

A joint chance-constrained programming approach for call center workforce scheduling under uncertain call arrival forecasts

We study the call center shift scheduling problem under uncertain demand forecasts.Forecasting errors are seen as independent normally distributed random variables.The resulting stochastic problem is modeled as a joint chance-constrained program.A mixed-integer linear programming based solution approach is proposed.Numerical results based on a real case study and managerial insights are provided. We consider a workforce management problem arising in call centers, namely the shift-scheduling problem. It consists in determining the number of agents to be assigned to a set of predefined shifts so as to optimize the trade-off between manpower cost and customer quality of service. We focus on explicitly taking into account in the shift-scheduling problem the uncertainties in the future call arrival rates forecasts. We model them as independent random variables following a continuous probability distribution. The resulting stochastic optimization problem is handled as a joint chance-constrained program and is reformulated as an equivalent large-size mixed-integer linear program. One key point of the proposed solution approach is that this reformulation is achieved without resorting to a scenario generation procedure to discretize the continuous probability distributions. Our computational results show that the proposed approach can efficiently solve real-size instances of the problem, enabling us to draw some useful managerial insights on the underlying risk-cost trade-off.

A MILP model and heuristic approach for facility location under multiple operational constraints

We study a multi-period facility location problem featuring many realistic constraints.We use a 2-phase solution approach: clustering of customers then facility location.Clustering aims at evaluating the average transport costs form facilities to customers.Exact and heuristic solution methods are tested using real life data.Insights related to multi-period modeling are discussed using the numerical results. In the present work, we study a multi-period facility location problem featuring many realistic constraints. In order to take into account vehicle routing from distribution centers to customers while maintaining a manageable size of the optimization problem, we develop a two-phase solution approach. In the first phase, the average distances and costs of transport from distribution centers to customers are evaluated using an exact clustering procedure based on a set-partitioning formulation. These costs serve as input to the facility location problem in the second phase, which is formulated as a mixed integer linear program and solved using a state-of-the art commercial solver. Many numerical experiments using real life data from the automotive industry are carried out in order to derive some insights related to multi-period modeling. We first show that in our case study, using static assignment decisions is better for the company as the corresponding operational benefit outweighs the additional cost to be incurred. We then compare the outputs of the multi-period model with those of its single-period counterpart. Finally, to cope with the computational difficulties encountered during the numerical experiments, we propose a linear relaxation based heuristic to solve larger instances of the problem. The heuristic method provides good quality solutions while significantly improving computation times.

Multi-product valid inequalities for the discrete lot-sizing and scheduling problem

We consider a problem arising in the context of industrial production planning, namely the multi-product discrete lot-sizing and scheduling problem with sequence-dependent changeover costs. We aim at developing an exact solution approach based on a Cut & Branch procedure for this combinatorial optimization problem. To achieve this, we propose a new family of multi-product valid inequalities which corresponds to taking into account the conflicts between different products simultaneously requiring production on the resource. We then present both an exact and a heuristic separation algorithm which form the basis of a cutting-plane generation algorithm. We finally discuss computational results which confirm the practical usefulness of the proposed inequalities at strengthening the MILP formulation and at reducing the overall computation time.

An evaluation of semidefinite programming based approaches for discrete lot-sizing problems☆

The present work is intended as a first step towards applying semidefinite programming models and tools to discrete lot-sizing problems including sequence-dependent changeover costs and times. Such problems can be formulated as quadratically constrained quadratic binary programs. We investigate several semidefinite relaxations by combining known reformulation techniques recently proposed for generic quadratic binary problems with problem-specific strengthening procedures developed for lot-sizing problems. Our computational results show that the semidefinite relaxations consistently provide lower bounds of significantly improved quality as compared with those provided by the best previously published linear relaxations. In particular, the gap between the semidefinite relaxation and the optimal integer solution value can be closed for a significant proportion of the small-size instances, thus avoiding to resort to a tree search procedure. The reported computation times are significant. However improvements in SDP technology can still be expected in the future, making SDP based approaches to discrete lot-sizing more competitive.

Distributionally Robust Optimization for Scheduling Problem in Call Centers with Uncertain Forecasts

This paper deals with the staffing and scheduling problem in call centers. We consider that the call arrival rates are subject to uncertainty and are following independent unknown continuous probability distributions. We assume that we only know the first and second moments of the distribution and thus propose to model this stochastic optimization problem as a distributionally robust program with joint chance constraints. Moreover, the risk level is dynamically shared throughout the entire scheduling horizon during the optimization process. We propose a deterministic equivalent of the problem and solve linear approximations of the Right-Hand Side of the program to provide upper and lower bounds of the optimal solution. We applied our approach on a real-life instance and give numerical results. Finally, we showed the practical interest of this approach compared to a stochastic approach in which the choice of the distribution is incorrect.

A joint chance-constrained programming approach for the single-item capacitated lot-sizing problem with stochastic demand

We study the single-item single-resource capacitated lot-sizing problem with stochastic demand. We propose to formulate this stochastic optimization problem as a joint chance-constrained program in which the probability that an inventory shortage occurs during the planning horizon is limited to a maximum acceptable risk level. We investigate the development of a new approximate solution method which can be seen as an extension of the previously published sample approximation approach. The proposed method relies on a Monte Carlo sampling of the random variables representing the demand in all planning periods except the first one. Provided there is no dependence between the demand in the first period and the demand in the later periods, this partial sampling results in the formulation of a chance-constrained program featuring a series of joint chance constraints. Each of these constraints involves a single random variable and defines a feasible set for which a conservative convex approximation can be quite easily built. Contrary to the sample approximation approach, the partial sample approximation leads to the formulation of a deterministic mixed-integer linear problem having the same number of binary variables as the original stochastic problem. Our computational results show that the proposed method is more efficient at finding feasible solutions of the original stochastic problem than the sample approximation method and that these solutions are less costly than the ones provided by the Bonferroni conservative approximation. Moreover, the computation time is significantly shorter than the one needed for the sample approximation method.

Scheduling Problem in Call Centers with Uncertain Arrival Rates Forecasts - A Distributionally Robust Approach

We focus on the staffing and shift-scheduling problem in call centers. We consider that the call arrival rates are subject to uncertainty and are following unknown continuous probability distributions. We assume that we only know the first and second moments of the distribution. We propose to model this stochastic optimization problem as a distributionally robust program with joint chance constraints. We consider a dynamic sharing out of the risk throughout the entire scheduling horizon. We propose a deterministic equivalent of the problem and solve linear approximations of the program to provide upper and lower bounds of the optimal solution. We applied our approach on a real-life instance and give numerical results.

Comparison of Stochastic Programming Approaches for Staffing and Scheduling Call Centers with Uncertain Demand Forecasts

We consider the staffing and shift-scheduling problems in call centers and propose a solution in one step. It consists in determining the minimum-cost number of agents to be assigned to each shift of the scheduling horizon so as to reach the required customer quality of service. We assume that the mean call arrival rate in each period of the horizon is a random variable following a continuous distribution. We model the resulting optimization problem as a stochastic program involving joint probabilistic constraints. We propose a solution approach based on linear approximations to provide approximate solutions of the problem. We finally compare them with other approaches and give numerical results carried out on a real-life instance. These results show that the proposed approach compares well with previously published approaches both in terms of risk management and cost minimization.