【算法项目】Capacitated Facility Location Problem

项目地址：https://github.com/GostBop/CFLP

Capacitated Facility Location Problem

问题描述

Suppose there are n facilities and m customers. We wish to choose:

1. which of the n facilities to open
2. the assignment of customers to facilities

Note：

1. The objective is to minimize the sum of the opening cost and the assignment cost.
2. The total demand assigned to a facility must not exceed its capacity.

问题建模：

Let
$I$={1,…, $I$}：the set of $facilities$
$J$={1,…, $J$}：the set of $customers$
$d_j$：$customer$ $j$ 's $Demand$，$j\in J$
$V_i$：$facility$ $i$ 's $Capacity$，$i\in I$
$C_i$：$facility$ $i$ 's $OpeningCost$，$i\in I$
$A_{ij}$: $facility$ $i$ 's $AssignmentCost$ for $customer$ $j$，$i\in I$，$j\in J$

For every $facility$ $i$，$i\in I$， define the following notations：
$$y_i=\{\begin{array}{ll}1& if.facility.i.is.opened\\ 0& otherwise\end{array}$$

For every $facility$ $i$，$i\in I$；$customer$ $j$，$j\in J$；
$$x_{ij}=\{\begin{array}{ll}1& if.facility.i.is.assigned.for.customer.j\\ 0& otherwise\end{array}$$

Then the Capacitated Facility Location Problem may be written：
∑ j ∈ J d j x i j ≤ V i y i ， i ∈ I \sum_{j∈J}d_jx_{ij} \leq V_iy_i，i∈I