The approach used to solve the network design problem is based on the Benders decomposition method where the subproblem is a mixed integer programming problem . 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 cut is generated by looking for the smallest stable transport cost for this configuration. The problem of finding the smallest stable transportation cost is itself solved by bender decomposition where the main problem searches for the worst demand for the fixed configuration and the subproblems generating the shear are simple flow problems using fixed configurations and demands. This section first presents some useful definitions, then successively proposes formulations for the flow subproblem, the stable transportation cost subproblem, and finally the global design problem. The distribution network contains factories, warehouses and customer areas. The problem, in every period, is to get the manufactured products to pass through the warehouses to the customers. The following definitions will be useful. The product flow is defined by the following two sets of variables. The quantity of product that passes from the factory to the warehouse. Since we next want to find a demand that maximizes the cost of the flow problem, it is useful to consider the dual of the previous problem to have a maximization objective function. where $lambda$, $alpha$, $mu$ and $sigma$ are, respectively, the dual variables of the constraints ef{lambda}, ef{alfa}, ef{mu} and ef{sigma}. Let us 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 that network. A binary variable $o_j$ determines whether the warehouse $j$ is open or not. To ensure that there is always enough space in the 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. Let $Delta$ be the finite set of possible values ​​of the binary variables $delta^+, delta^-, w, f$ and let $h_r(delta) = (sigma_{r,delta},z_ {r,delta}, alpha_{r,delta},mu_{r,delta}), r = 1 ldots m_delta$ the set of extreme points of the problem ( ef{WorstBin}) with the value of the binary variable fixed. Since the ef{Design1} problem is convex with respect to the $gamma$ variables we can use a Benders decomposition approach to solve the following network design problem.