linopy: Optimization with array-like variables and constraints

        Linear
        Integer
        Non-linear
        Optimization in
        PYthon

linopy is an open-source python package that facilitates optimization with real world data. It builds a bridge between data analysis packages like xarray & pandas and problem solvers like cbc, gurobi (see the full list below). Linopy supports Linear, Integer, Mixed-Integer and Quadratic Programming while aiming to make linear programming in Python easy, highly-flexible and performant.

Benchmarks

linopy is designed to be fast and efficient. The following benchmark compares the performance of linopy with the alternative popular optimization packages.

Main features

linopy is heavily based on xarray which allows for many flexible data-handling features:

• Define (arrays of) continuous or binary variables with coordinates, e.g. time, consumers, etc.
• Apply arithmetic operations on the variables like adding, substracting, multiplying with all the broadcasting potentials of xarray
• Apply arithmetic operations on the linear expressions (combination of variables)
• Group terms of a linear expression by coordinates
• Get insight into the clear and transparent data model
• Modify and delete assigned variables and constraints on the fly
• Use lazy operations for large linear programs with dask
• Choose from different commercial and non-commercial solvers
• Fast import and export a linear model using xarray's netcdf IO

Installation

So far linopy is available on the PyPI repository

pip install linopy

or on conda-forge

conda install -c conda-forge linopy

In a Nutshell

Linopy aims to make optimization programs transparent and flexible. To illustrate its usage, let's consider a scenario where we aim to minimize the cost of buying apples and bananas over a week, subject to daily and weekly vitamin intake constraints.

>>> import pandas as pd
>>> import linopy

>>> m = linopy.Model()

>>> days = pd.Index(['Mon', 'Tue', 'Wed', 'Thu', 'Fri'], name='day')
>>> apples = m.add_variables(lower=0, name='apples', coords=[days])
>>> bananas = m.add_variables(lower=0, name='bananas', coords=[days])
>>> apples

Variable (day: 5)
-----------------
[Mon]: apples[Mon] ∈ [0, inf]
[Tue]: apples[Tue] ∈ [0, inf]
[Wed]: apples[Wed] ∈ [0, inf]
[Thu]: apples[Thu] ∈ [0, inf]
[Fri]: apples[Fri] ∈ [0, inf]

Add daily vitamin constraints

>>> m.add_constraints(3 * apples + 2 * bananas >= 8, name='daily_vitamins')

Constraint `daily_vitamins` (day: 5):
-------------------------------------
[Mon]: +3 apples[Mon] + 2 bananas[Mon] ≥ 8
[Tue]: +3 apples[Tue] + 2 bananas[Tue] ≥ 8
[Wed]: +3 apples[Wed] + 2 bananas[Wed] ≥ 8
[Thu]: +3 apples[Thu] + 2 bananas[Thu] ≥ 8
[Fri]: +3 apples[Fri] + 2 bananas[Fri] ≥ 8

Add weekly vitamin constraint

>>> m.add_constraints((3 * apples + 2 * bananas).sum() >= 50, name='weekly_vitamins')

Constraint `weekly_vitamins`
----------------------------
+3 apples[Mon] + 2 bananas[Mon] + 3 apples[Tue] ... +2 bananas[Thu] + 3 apples[Fri] + 2 bananas[Fri] ≥ 50

Define the prices of apples and bananas and the objective function

>>> apple_price = [1, 1.5, 1, 2, 1]
>>> banana_price = [1, 1, 0.5, 1, 0.5]
>>> m.objective = apple_price * apples + banana_price * bananas

Finally, we can solve the problem and get the optimal solution:

>>> m.solve()
>>> m.objective.value

17.166

... and display the solution as a pandas DataFrame

>>> m.solution.to_pandas()

        apples  bananas
day
Mon    2.667      0
Tue    0          4
Wed    0          9
Thu    0          4
Fri    0          4

Supported solvers

linopy supports the following solvers

• Cbc
• GLPK
• HiGHS
• Gurobi
• Xpress
• Cplex
• MOSEK
• COPT

Note that these do have to be installed by the user separately.

Citing Linopy

If you use Linopy in your research, please cite the following paper:

• Hofmann, F., (2023). Linopy: Linear optimization with n-dimensional labeled variables. Journal of Open Source Software, 8(84), 4823, https://doi.org/10.21105/joss.04823

A BibTeX entry for LaTeX users is

@article{Hofmann2023,
    doi = {10.21105/joss.04823},
    url = {https://doi.org/10.21105/joss.04823},
    year = {2023}, publisher = {The Open Journal},
    volume = {8},
    number = {84},
    pages = {4823},
    author = {Fabian Hofmann},
    title = {Linopy: Linear optimization with n-dimensional labeled variables},
    journal = {Journal of Open Source Software}
}

License

Copyright 2021 Fabian Hofmann

This package is published under MIT license. See LICENSE.txt for details.