# Essay on Benders' Decomposition Approach To Solve Network Design Problem

# Essay on Benders' Decomposition Approach To Solve Network Design Problem

**Length:** 1135 words (3.2 double-spaced pages)

**Rating:** Strong Essays

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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}).

...

... middle of paper ...

...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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