# Benders' Decomposition Approach To Solve Network Design Problem

1048 words - 4 pages

The approach used to solve the network design problem is based on a Benders' decomposition method where the sub-problem is a mixed integer programming problem. The master problem consists of choosing the best configuration $Q$ given the current set of constraints, where $Q$ is the warehouses capacity vector. Once this configuration is found, a cut is generated by searching for the smallest stable transportation cost for this configuration. The problem of finding the smallest stable transportation cost is itself solve by bender decomposition where the master problem search for the worst demand for the fixed configuration and the cut generating sub-problems are simple flow problems using fixed configuration and demands. This section first present some useful definitions, then successively proposes the formulations for the flow sub-problem, the stable transportation cost sub-problem and finally the global design problem.

The distribution network contains mills, warehouses and customers zones. The problem, for every period, consist of transiting manufactured products through warehouses to the customers. The following definitions will be useful.
The flow of products is defined by the two following set of variables.
The quantity of product transiting from the mill 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 constraints \ref{lambda}, \ref{alpha}, \ref{mu}, and \ref{sigma}. We call $\Phi(Q,d)$ the optimal value of the objective function for the linear program (\ref{FlowDual}).

Now we consider, given a fixed $Q$, the problem of finding the value of the demand $d$ that will maximize $\Phi(Q,d)$. We want the total demand to remain constant, that is every unit of demand lost by a customer is gained by another customer, also the worst that could happen is that at most $\Gamma$ unit of demands can be subtracted from the base demand for all customers and no customer will more than double is demand. Only the total demand of a customer change, the relative proportion of this demand across all products remains the same for every customer. If we define $\delta^{+}_{k}$ and $\delta^{-}_{k}$ to be respectively the positive and negative change to the demand of customer $k$ we can modify the dual of the flow sub-problem (\ref{FlowDual}) to incorporate this varying demand, but doing so results in a non linear objective function:

One way around non linear objective function is to use a mixed integer formulation similar to the one proposed in \citet{thiele2009}.
It can be shown that there always exist an optimal solution where, except for two customers, every customer either double is demand, don't change is demand, or have null demand. The two exceptions, named respectively the positive and negative...

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