Lecture IV - Modelling with JuMP

Applied Optimization with Julia

Dr. Tobias Vlćek

University of Hamburg

Quick Recap from last Week

Goals for Today

After this lecture, you will:

  • Have refreshed functions, packages, DataFrames, IO, and plots
  • Know the difference between a model and a solver
  • Have seen the transport problem from the first lecture solved in JuMP
  • Be ready to build your first own models in the tutorials

Functions

  • Functions are reusable blocks of code
  • Define functions using the function keyword
  • Functions can take arguments and return values
  • Use return to specify the output of a function
  • This week: functions help you to structure your models

You can create anonymous functions using the -> syntax for quick, one-off operations.

Packages

  • Packages extend Julia’s functionality
  • Use using Pkg to access package management
  • Install packages with Pkg.add("PackageName")
  • Or press ] in the REPL to enter package mode and type add PackageName
  • Import packages with using PackageName or import PackageName
  • This week: you will install two new packages, JuMP and HiGHS

DataFrames

  • DataFrames are used for working with tabular data
  • Create DataFrames using the DataFrame constructor
  • Access columns using dot notation or square brackets
  • Perform operations on columns and rows
  • This week: DataFrames can hold the data of your models

Use describe(df) to get a quick summary of your DataFrame.

Input/Output (IO)

  • IO operations allow reading from and writing to files
  • Reading and writing CSV files can be done with the CSV package
  • Use CSV.read("file.csv", DataFrame) to read a CSV file into a DataFrame
  • Use CSV.write() to write a DataFrame to a CSV file
  • This week: this is how you load model data from files

The DataFrame is a required second argument of CSV.read() — it tells the function into which structure to read the file.

Plots

  • Plotting in Julia is done through packages like Plots.jl
  • Create basic plots with functions like plot(), scatter(), bar()
  • Customize plots with attributes like title, xlabel, ylabel
  • This week: plots help you to inspect data and results

Explore different plot types; later, different backends offer other output formats and interactivity.

Solutions from last Week

  • The solutions to the tutorials are made available at the end of each week
  • You can access them in the project folder on GitHub
  • Click on the GitHub icon (the Octocat) at the bottom right of the page

You can ask questions anytime in class or via email!

Modelling with JuMP

What is JuMP?

Question: How do we get a model from paper into the computer?

  • JuMP is a package for modelling optimization problems
  • We write variables, objective, and constraints almost like the math
  • JuMP itself does not solve the model — it hands it to a solver
  • Ours is HiGHS, a free and open-source solver

Model and solver are separate: JuMP describes what to optimize, HiGHS figures out how to solve it.

Remember the Solar Panels?

In the first lecture, we formulated the transport problem:

\[ \begin{aligned} \text{Minimize} \quad F &= \sum_{i \in \mathcal{I}} \sum_{j \in \mathcal{J}} c_{i,j} \times X_{i,j} \\ \text{subject to:} \quad &\sum_{j \in \mathcal{J}} X_{i,j} \leq a_i \quad &&\forall i \in \mathcal{I} \\ &\sum_{i \in \mathcal{I}} X_{i,j} = b_j \quad &&\forall j \in \mathcal{J} \\ &X_{i,j} \geq 0 \quad &&\forall i \in \mathcal{I}, \forall j \in \mathcal{J} \end{aligned} \]

Let’s now solve it — live in JuMP!

Step 1: Data, Model, and Variables

using JuMP, HiGHS
c = [5010 4640 1980; 7120 1710 6430] # costs per truckload
a = [34, 41]                         # available supply
b = [21, 17, 29]                     # customer demand
transport = Model(HiGHS.Optimizer)
set_silent(transport)                # hide the solver log
@variable(transport, X[1:2, 1:3] >= 0);
  • First the data, then the model with its solver, then the variables \(X_{i,j} \geq 0\)

Step 2: Objective and Constraints

@objective(transport, Min,
    sum(c[i,j] * X[i,j] for i in 1:2, j in 1:3))
@constraint(transport, [i in 1:2],
    sum(X[i,j] for j in 1:3) <= a[i])
@constraint(transport, [j in 1:3],
    sum(X[i,j] for i in 1:2) == b[j]);
  • Compare this with the mathematical model: each line of math becomes one block of code

Step 3: Solve the Model

optimize!(transport)
println("Minimal costs: ", objective_value(transport), " Euro")
Minimal costs: 225460.0 Euro
  • Exactly the optimal solution from the first lecture!

Don’t worry if not every line is clear yet — this week’s tutorials build up this pattern step by step.

Five Tutorials for this Week

Topics of the Tutorials

  • JuMP: Learn how to use JuMP to define optimization problems
  • Variable Bounds: Learn how to set variable bounds
  • Constraints: Learn how to add constraints to your model
  • Advanced Modeling: Learn how to model more complex problems
  • Transport Problem: Learn how to solve the transport problem from the first lecture

Get Started with the Tutorials

  • Download this week’s tutorials and start with the first one
  • Remember, you can ask questions anytime!

And that’s it for this lecture!

The remaining time we will already start working on this week’s tutorials. From next week on, we will work on different optimization problems and topics that we will address together in the course.

Literature

Literature

For more interesting literature to learn more about Julia, take a look at the literature list of this course.