Network models with node capacities

Min-cost flow network model

A standard min-cost flow network model, formulated as a linear programming model, can look like:

Model 1: standard min-cost flow network model
\[\begin{align}\min& \sum_{(i,j)\in A}\color{darkblue}c_{i,j}\cdot\color{darkred}f_{i,j}\\ & \sum_{j|(j,i)\in A}\color{darkred}f_{j,i} + \color{darkblue}{\mathit{supply}}_i = \sum_{j|(i,j)\in A}\color{darkred}f_{i,j} + \color{darkblue}{\mathit{demand}}_i && \forall i \\ & \color{darkred}f_{i,j} \in [0,\color{darkblue}{\mathit{capacity}}_{i,j}] \end{align}\]

Solving as network with lowerbounds

In [1], we looked at the following problem:

Mathematical Model
\[ \begin{align} \min& \sum_{i,j} \color{darkblue}a_{i,j} \cdot \color{darkred}x_{i,j} \\ & \sum_j \color{darkblue}a_{i,j}\cdot \color{darkred}x_{i,j} \ge \color{darkblue}r_i && \forall i \\ & \sum_i \color{darkblue}a_{i,j}\cdot \color{darkred}x_{i,j} \ge \color{darkblue}c_j && \forall j \\ & \color{darkred}x_{i,j} \in \{0,1\} \end{align}\]

A large MIP model that should be solved as LP: the root node

In this post, I want to discuss an observation about the root node when solving a MIP model.

Problem Description

The model I want to tackle here is from [1]:

We have a large matrix \(\color{darkblue}A=\color{darkblue}a_{i,j}\) with values 0 and 1. In addition, we have a minimum on the row and column totals. These are called \(\color{darkblue}r_i\) and \(\color{darkblue}c_j\). The goal is to remove as many 1's in the matrix \(\color{darkblue}A\) subject to these minimum row and column totals.

Networks not always need an extra source and sink node

Convert standard assignment problem to a network problem

A standard assignment problem formulated as an LP/MIP looks like:

 

Assignment problem
\[\begin{align}\min&\sum_{i,j} \color{darkblue}c_{i,j}\cdot\color{darkred}x_{i,j}\\ & \sum_j \color{darkred}x_{i,j}=1 && \forall i \\ & \sum_i \color{darkred}x_{i,j}=1 && \forall j \\ & \color{darkred}x_{i,j} \in \{0,1\} \end{align}\]