Essay on Benders' Decomposition Approach To Solve Network Design Problem

Essay on Benders' Decomposition Approach To Solve Network Design Problem

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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}).
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...st $r_j$, the variable inventory cost $i_j$ and the transportation cost $\Phi(Q)$ knowing that we will have to face the worst demand for that network. A binary variable $o_j$ determines whether or not warehouse $j$ is open.
To ensure that there is always enough space in the warehouses to fulfill 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 value of the binary variables $\delta^+, \delta^-, w, f$ and let $h_r(\delta) = (\sigma_{r,\delta},z_{r,\delta},\alpha_{r,\delta},\mu_{r,\delta}), r = 1 \ldots m_\delta$ be the set of extreme points of the problem (\ref{WorstBin}) with the value of the binary variable fixed. Since problem \ref{Design1} is convex with respect to variables $\gamma$ we can use a Benders' decomposition approach to solve the following network design problem.

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