The approach used to solve the network design problem is based on a Benders decomposition method where the subproblem is a problem mixed integer programming. The main problem is to choose the best configuration $Q$ given the current set of constraints, where $Q$ is the warehouse capacity vector. Once this configuration is found, a discount is generated by finding the lowest stable transportation cost for this configuration. The problem of finding the smallest stable transportation cost is itself solved by a bender decomposition where the main problem searches for the worst demand for the fixed configuration and the subproblems generating outages are simple flow problems using a configuration and fixed demands. This section first presents some useful definitions, then successively proposes formulations for the flow subproblem, the stable transportation cost subproblem and finally the overall design problem. The distribution network contains factories, warehouses and customer areas. The problem in each period is moving manufactured goods through warehouses to customers. The following definitions will be helpful. The product flow is defined by the following two sets of variables. The quantity of product moving from the factory to the warehouse. Since later we want to find a demand that maximizes the cost of the flow problem, it is useful to consider the dual of the previous problem in order to have a maximization objective function. where $lambda$, $alpha$, $mu$ and $sigma$ are, respectively, the dual variables of the constraints ef{lambda}, ef{alpha}, ef{mu} and ef{sigma}. We call $Phi(Q,d)$ the optimal value of the objective function for the linear program ( ef{FlowDual}). ...... middle of paper ......st $r_j$, the variable inventory cost $i_j$ and the transportation cost $Phi(Q)$ knowing that we will face the worst demand for this network. A binary variable $o_j$ determines whether warehouse $j$ is open or not. To ensure that there is always enough space in warehouses to meet any demand, we require that $sum_{j in W} Q_j geq sum_{s in S} d_s$. The design problem is as follows: Let $Delta$ be the finite set of possible values ​​of the binary variables $delta^+, delta^-, w, f$ and let $h_r(delta) = (sigma_{r,delta} ,z_ {r,delta},alpha_{r,delta},mu_{r,delta}), r = 1 ldots m_delta$ i.e. the set of extreme points of the problem (ef{WorstBin}) with the value of the variable fixed binary. Since the problem ef{Design1} is convex with respect to the $gamma$ variables, we can use a Benders decomposition approach to solve the following network design problem.