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
\[\begin{align}\min& \sum_{(i,j)\in A}\color{darkblue}c_{i,j}\cdot\color{darkred}f_{i,j}\\ & \sum_{j|(j,i)\in A}\color{darkred}f_{j,i} + \color{darkblue}{\mathit{supply}}_i = \sum_{j|(i,j)\in A}\color{darkred}f_{i,j} + \color{darkblue}{\mathit{demand}}_i && \forall i \\ & \color{darkred}f_{i,j} \in [0,\color{darkblue}{\mathit{capacity}}_{i,j}] \end{align}\]

Sorting using a MIP model

This is not, per se, very useful, but sorting a parameter inside a MIP model is not very easy for MIP solvers. Obviously, if you need a sorted parameter in the model, it is better to use a sorting algorithm. But useless models can still be interesting.

I use: 

    Input: a 1-dimensional parameter with values.
    Output: a 1-dimensional variable with the values sorted in ascending order.

We can implement this with a permutation matrix \(\color{darkred}X\), which is a permuted identity matrix. In a MIP context, this becomes a binary variable with some assignment constraints.

MIP Model for sorting \(p_i\)
\[\begin{align}\min\>&\color{darkred}z=0\\& \sum_i \color{darkred}x_{i,j} = 1&&\forall j\\ & \sum_j \color{darkred}x_{i,j} = 1&&\forall i\\ & \color{darkred}y_i = \sum_j \color{darkred}x_{i,j}\cdot\color{darkblue}p_j\\& \color{darkred}y_i \ge \color{darkred}y_{i-1}\\ & \color{darkred}x_{i,j} \in \{0,1\}\end{align}\]

PuLP surprises

Formulating optimization models inside traditional programming languages such as Python is very popular. The main tool the developers use to make this possible is operator overloading. There are cases, where we can write code that looks somewhat reasonable, is accepted and processed without any warning or error messages, but is total nonsense. It is rather difficult with this approach to make things airtight. Especially error handling. In [1], we see a good example. I have created a small fragment here that illustrates the problem.

Non-convex Quadratic Integer Programming

Here, I want to revisit a particular model from [1]:

Model 3: Quadratic Preemptive Model

\[\begin{align}\max\>&\color{darkred}z_{model3}=\sum_p \color{darkred}z_p \\ & \color{darkred}z_p = \sum_{p',g} \color{darkblue}{\mathit{pref}}_{p,p'}\cdot \color{darkred}{\mathit{assign}}_{p,g}\cdot\color{darkred}{\mathit{assign}}_{p',g} \\ & \color{darkblue}z_{model2}^* \le \color{darkred}z_p & \forall p\\ & \sum_g \color{darkred}{\mathit{assign}}_{p,g} = 1 & \forall p \\ & \sum_p \color{darkred}{\mathit{assign}}_{p,g} = \color{darkblue}{\mathit{groupSize}} & \forall g \\& \color{darkred}{\mathit{assign}}_{p,g} \in \{0,1\}\end{align}\]

Equity in optimization models

In optimization models, we often use an aggregate measure in the objective function, such as total profit, the sum of tardiness of jobs, and countrywide GDP. This can lead to particularly bad results for some individuals or groups.

Here is an example I have used on several occasions. 

Problem Statement

We have \(P\) persons. They must be assigned to \(M\) groups or teams. For simplicity, we can assume \(n\) is a multiple of \(m\), and the group size is \[\frac{N}{M}\] Each person \(p_1\) specifies some preferences to be placed in the same group as a person \(p_2\). A negative preference can be used to indicate that I prefer to be in a different group. Find an optimal assignment taking into account these preferences. 

GAMS 48 tests

Some minor quibbles. 

gdx2sqlite

The latest version of GAMS contains a replacement of gdx2sqlite. This dumps a GDX file into a SQLite database. It is a tool I use a lot. Here is a comparison using the indus89 model in the GAMS model library:

Prevent Loops in GAMS

 

This book [1] on DEA models has an accompanying website with all the GAMS models [2]. 

Of course, I'll be doing some nitpicking on the GAMS code. 

CSV readers mutilating my data

R and CSV files

When I deal with regional codes such as FIPS[1] and HUC[2], CSV file readers often mutilate my regions. Here is an example in R: 

Solving DEA Models with GAMS

Data Envelopment Analysis (DEA) models are somewhat special. They typically consist of small LPs, of which a whole bunch have to be solved. The CCR formulation (after [1]), for the \(i\)-th DMU (Decision Making Unit), can be stated as [2]:

CCR LP Model

\[\begin{align} \max \>& \color{darkred}{\mathit{efficiency}}_i=\sum_{\mathit{outp}} \color{darkred}u_{{\mathit{outp}}} \cdot \color{darkblue}y_{i,{\mathit{outp}}} \\ & \sum_{\mathit{inp}} \color{darkred}v_{{\mathit{inp}}} \cdot \color{darkblue}x_{i,{\mathit{inp}}} = 1 \\ & \sum_{\mathit{outp}} \color{darkred}u_{{\mathit{outp}}} \cdot \color{darkblue}y_{j,{\mathit{outp}}} \le \color{darkred}v_{{\mathit{inp}}} \cdot \color{darkblue}x_{j,{\mathit{inp}}} && \forall j \\ & \color{darkred}u_{{\mathit{outp}}} \ge 0, \color{darkred}v_{{\mathit{inp}}} \ge 0 \end{align}\]

Multiple Solutions in Minimum Spanning Tree example

In [1], I discussed some LP and MIP formulations for the Minimum Spanning Tree (MST) problem. 

Minimum Spanning Tree visualized through Google Maps

Here, I focus on two formulations: a multicommodity network approach (this can be solved as a large LP) and a MIP formulation based on techniques we know from the Traveling Salesman Problem (TSP). The main issue I want to discuss is the presence of multiple optimal solutions.