Model and Solve Mixed Integer Linear Programs.

Mixed integer linear programming in R

OMPR (Optimization Modeling Package) is a DSL to model and solve Mixed Integer Linear Programs. It is inspired by the excellent Jump project in Julia.

Here are some problems you could solve with this package:

• What is the cost minimal way to visit a set of clients and return home afterwards?
• What is the optimal conference time table subject to certain constraints (e.g. availability of a projector)?
• Sudokus

The Wikipedia article gives a good starting point if you would like to learn more about the topic.

I am always happy to get bug reports or feedback.

Install

CRAN

install.packages("ompr")
install.packages("ompr.roi")

Development version

To install the current development version use devtools:

remotes::install_github("dirkschumacher/ompr")
remotes::install_github("dirkschumacher/ompr.roi")

Available solver bindings

• ompr.roi - Bindings to ROI (GLPK, Symphony, CPLEX etc.)

A simple example:

suppressPackageStartupMessages(library(dplyr, quietly = TRUE)) 
suppressPackageStartupMessages(library(ROI))
library(ROI.plugin.glpk)
library(ompr)
library(ompr.roi)

result <- MIPModel() |>
  add_variable(x, type = "integer") |>
  add_variable(y, type = "continuous", lb = 0) |>
  set_bounds(x, lb = 0) |>
  set_objective(x + y, "max") |>
  add_constraint(x + y <= 11.25) |>
  solve_model(with_ROI(solver = "glpk"))
get_solution(result, x)
#>  x 
#> 11
get_solution(result, y)
#>    y 
#> 0.25

API

These functions currently form the public API. More detailed docs can be found in the package function docs or on the website

DSL

• MIPModel() create an empty mixed integer linear model (the old way)
• add_variable() adds variables to a model
• set_objective() sets the objective function of a model
• set_bounds() sets bounds of variables
• add_constraint() add constraints
• solve_model() solves a model with a given solver
• get_solution() returns the column solution (primal or dual) of a solved model for a given variable or group of variables
• get_row_duals() returns the row duals of a solution (only if it is an LP)
• get_column_duals() returns the column duals of a solution (only if it is an LP)

Backends

There are currently two backends. A backend is the function that initializes an empty model.

• MIPModel() is the standard MILP Model.
• MILPModel() is another backend specifically optimized for linear models and is often faster than MIPModel(). It has different semantics, as it is vectorized. Currently experimental and might be deprecated in the future.

Solvers

Solvers are in different packages. ompr.ROI uses the ROI package which offers support for all kinds of solvers.

• with_ROI(solver = "glpk") solve the model with GLPK. Install ROI.plugin.glpk
• with_ROI(solver = "symphony") solve the model with Symphony. Install ROI.plugin.symphony
• with_ROI(solver = "cplex") solve the model with CPLEX. Install ROI.plugin.cplex
• … See the ROI package for more plugins.

Further Examples

Please take a look at the docs for bigger examples.

Knapsack

max_capacity <- 5
n <- 10
set.seed(1234)
weights <- runif(n, max = max_capacity)
MIPModel() |>
  add_variable(x[i], i = 1:n, type = "binary") |>
  set_objective(sum_over(weights[i] * x[i], i = 1:n), "max") |>
  add_constraint(sum_over(weights[i] * x[i], i = 1:n) <= max_capacity) |>
  solve_model(with_ROI(solver = "glpk")) |>
  get_solution(x[i]) |>
  filter(value > 0)
#>   variable i value
#> 1        x 1     1
#> 2        x 6     1
#> 3        x 7     1
#> 4        x 8     1

Bin Packing

An example of a more difficult model solved by GLPK

max_bins <- 10
bin_size <- 3
n <- 10
weights <- runif(n, max = bin_size)
MIPModel() |>
  add_variable(y[i], i = 1:max_bins, type = "binary") |>
  add_variable(x[i, j], i = 1:max_bins, j = 1:n, type = "binary") |>
  set_objective(sum_over(y[i], i = 1:max_bins), "min") |>
  add_constraint(sum_over(weights[j] * x[i, j], j = 1:n) <= y[i] * bin_size, i = 1:max_bins) |>
  add_constraint(sum_over(x[i, j], i = 1:max_bins) == 1, j = 1:n) |>
  solve_model(with_ROI(solver = "glpk", verbose = TRUE)) |>
  get_solution(x[i, j]) |>
  filter(value > 0) |>
  arrange(i)
#> <SOLVER MSG>  ----
#> GLPK Simplex Optimizer, v4.65
#> 20 rows, 110 columns, 210 non-zeros
#>       0: obj =   0.000000000e+00 inf =   1.000e+01 (10)
#>      29: obj =   4.546337429e+00 inf =   0.000e+00 (0)
#> *    34: obj =   4.546337429e+00 inf =   0.000e+00 (0)
#> OPTIMAL LP SOLUTION FOUND
#> GLPK Integer Optimizer, v4.65
#> 20 rows, 110 columns, 210 non-zeros
#> 110 integer variables, all of which are binary
#> Integer optimization begins...
#> Long-step dual simplex will be used
#> +    34: mip =     not found yet >=              -inf        (1; 0)
#> +    62: >>>>>   5.000000000e+00 >=   5.000000000e+00   0.0% (13; 0)
#> +    62: mip =   5.000000000e+00 >=     tree is empty   0.0% (0; 25)
#> INTEGER OPTIMAL SOLUTION FOUND
#> <!SOLVER MSG> ----
#>    variable  i  j value
#> 1         x  1  2     1
#> 2         x  1  9     1
#> 3         x  1 10     1
#> 4         x  2  5     1
#> 5         x  2  7     1
#> 6         x  2  8     1
#> 7         x  3  6     1
#> 8         x  4  4     1
#> 9         x 10  1     1
#> 10        x 10  3     1

License

MIT

Contributing

Please post an issue first before sending a PR.

Please note that this project is released with a Contributor Code of Conduct. By participating in this project you agree to abide by its terms.

Related Projects

• CVXR - an excellent package for “object-oriented modeling language for convex optimization”. LP/MIP is a special case.
• ROML follows a similar approach, but it seems the package is still under initial development.