This article discribes the modelling techniques available in ompr
to make your life easier when developing a mixed integer programming model.
You can think of a MIP Model as a big constraint maxtrix and a set of vectors. But you can also think of it as a set of decision variables, an objective function and a number of constraints as equations/inequalities. ompr
implements the latter approach.
For example, Wikipedia describes the Knapsack problem like this: \[ \begin{equation*} \begin{array}{ll@{}ll} \text{max} & \displaystyle\sum\limits_{i=1}^{n} v_{i}x_{i} & &\\ \text{subject to}& \displaystyle\sum\limits_{i=1}^{n} w_{i}x_{i} \leq W, & &\\ & x_{i} \in \{0,1\}, &i=1 ,\ldots, n& \end{array} \end{equation*} \]
This is the ompr
equivalent:
n <- 10; W <- 2
v <- runif(n);w <- runif(n)
model <- MIPModel() %>%
add_variable(x[i], i = 1:n, type = "binary") %>%
set_objective(sum_expr(v[i] * x[i], i = 1:n)) %>%
add_constraint(sum_expr(w[i] * x[i], i = 1:n) <= W)
The overall idea is to use modern R idioms to construct models like the one above as readable as possible directly in R. ompr
will do the heavy lifting and transforms everything into matrices/vectors and pass it to your favorite solver.
Each function in ompr
creates immutable copies of the models. In addition the function interface has been designed to work with magrittr
pipes. You always start with an empty model and add components to it.
MIPModel() %>%
add_variable(x) %>%
set_objective(x) %>%
add_constraint(x <= 1)
## Mixed integer linear optimization problem
## Variables:
## Continuous: 1
## Integer: 0
## Binary: 0
## Model sense: maximize
## Constraints: 1
Variables can be of type continuous
, integer
or binary
.
MIPModel() %>%
add_variable(x, type = "integer") %>%
add_variable(y, type = "continuous") %>%
add_variable(z, type = "binary")
## Mixed integer linear optimization problem
## Variables:
## Continuous: 1
## Integer: 1
## Binary: 1
## No objective function.
## Constraints: 0
Variables can have lower and upper bounds.
MIPModel() %>%
add_variable(x, lb = 10) %>%
add_variable(y, lb = 5, ub = 10)
## Mixed integer linear optimization problem
## Variables:
## Continuous: 2
## Integer: 0
## Binary: 0
## No objective function.
## Constraints: 0
Often when you develop a complex model you work with indexed variables. This is an important concept ompr
supports.
MIPModel() %>%
add_variable(x[i], i = 1:10) %>% # creates 10 decision variables
set_objective(x[5]) %>%
add_constraint(x[5] <= 10)
## Mixed integer linear optimization problem
## Variables:
## Continuous: 10
## Integer: 0
## Binary: 0
## Model sense: maximize
## Constraints: 1
If you have indexed variables then you often want to sum over a subset of variables.
The following code creates a model with three decision variables \(x_1\), \(x_2\), \(x_3\). An objective function \(\sum_i x_i\) and one constraint \(\sum_i x_i \leq 10\).
MIPModel() %>%
add_variable(x[i], i = 1:3) %>%
set_objective(sum_expr(x[i], i = 1:3)) %>%
add_constraint(sum_expr(x[i], i = 1:3) <= 10)
## Mixed integer linear optimization problem
## Variables:
## Continuous: 3
## Integer: 0
## Binary: 0
## Model sense: maximize
## Constraints: 1
add_variable
, add_constraint
, set_bounds
, sum_expr
all support a common quantifier interface that also supports filter expression. A more complex example will show what that means.
MIPModel() %>%
# Create x_{i, j} variables for all combinations of i and j where
# i = 1:10 and j = 1:10.
add_variable(x[i, j], type = "binary", i = 1:10, j = 1:10) %>%
# add a y_i variable for all i between 1 and 10 with i mod 2 = 0
add_variable(y[i], type = "binary", i = 1:10, i %% 2 == 0) %>%
# we maximize all x_{i,j} where i = j + 1
set_objective(sum_expr(x[i, j], i = 1:10, j = 1:10, i == j + 1)) %>%
# for each i between 1 and 10 with i mod 2 = 0
# we add a constraint \sum_j x_{i,j}
add_constraint(sum_expr(x[i, j], j = 1:10) <= 1, i = 1:10, i %% 2 == 0) %>%
# of course you can leave out filters or add more than 1
add_constraint(sum_expr(x[i, j], j = 1:10) <= 2, i = 1:10)
## Mixed integer linear optimization problem
## Variables:
## Continuous: 0
## Integer: 0
## Binary: 105
## Model sense: maximize
## Constraints: 15
Imagine you want to model a matching problem with a single binary decision variable \(x_{i,j}\) that is \(1\) iff object \(i\) is matched to object \(j\). One constraint would be to allow matches only if \(i \neq j\). This can be modelled by a constraint or by selectively changing bounds on variables. The latter approach can be used by solvers to improve the solution process.
MIPModel() %>%
add_variable(x[i, j], i = 1:10, j = 1:10,
type = "integer", lb = 0, ub = 1) %>%
set_objective(sum_expr(x[i, j], i = 1:10, j = 1:10)) %>%
add_constraint(x[i, i] == 0, i = 1:10) %>%
# this sets the ub to 0 without adding new constraints
set_bounds(x[i, i], ub = 0, i = 1:10)
## Mixed integer linear optimization problem
## Variables:
## Continuous: 0
## Integer: 100
## Binary: 0
## Model sense: maximize
## Constraints: 10
Of course you will need external parameters for your models. You can reuse any variable defined in your R environment within the MIP Model.
n <- 5 # number of our variables
costs <- rpois(n, lambda = 3) # a cost vector
max_elements <- 3
MIPModel() %>%
add_variable(x[i], type = "binary", i = 1:n) %>%
set_objective(sum_expr(costs[i] * x[i], i = 1:n)) %>%
add_constraint(sum_expr(x[i], i = 1:n) <= max_elements)
## Mixed integer linear optimization problem
## Variables:
## Continuous: 0
## Integer: 0
## Binary: 5
## Model sense: maximize
## Constraints: 1
Once you have a model, you pass it to a solver and get back a solutions. The main interface to extract variable values from a solution is the function get_solution
. It returns a data.frame for indexed variable and thus makes it easy to subsequently use the value.
We use ROI
and GLPK
to solve it.
library(ROI)
## ROI.plugin.glpk: R Optimization Infrastructure
## Registered solver plugins: nlminb, cbc, glpk.
## Default solver: auto.
library(ROI.plugin.glpk)
library(ompr.roi)
set.seed(1)
n <- 5
weights <- matrix(rpois(n * n, 5), ncol = n, nrow = n)
result <- MIPModel() %>%
add_variable(x[i, j], i = 1:n, j = 1:n, type = "binary") %>%
set_objective(sum_expr(weights[i, j] * x[i, j], i = 1:n, j = 1:n)) %>%
add_constraint(sum_expr(x[i, j], j = 1:n) == 1, i = 1:n) %>%
solve_model(with_ROI("glpk", verbose = TRUE))
## <SOLVER MSG> ----
## GLPK Simplex Optimizer, v4.65
## 5 rows, 25 columns, 25 non-zeros
## 0: obj = -0.000000000e+00 inf = 5.000e+00 (5)
## 5: obj = 2.400000000e+01 inf = 0.000e+00 (0)
## * 14: obj = 4.400000000e+01 inf = 0.000e+00 (0)
## OPTIMAL LP SOLUTION FOUND
## GLPK Integer Optimizer, v4.65
## 5 rows, 25 columns, 25 non-zeros
## 25 integer variables, all of which are binary
## Integer optimization begins...
## Long-step dual simplex will be used
## + 14: mip = not found yet <= +inf (1; 0)
## + 14: >>>>> 4.400000000e+01 <= 4.400000000e+01 0.0% (1; 0)
## + 14: mip = 4.400000000e+01 <= tree is empty 0.0% (0; 1)
## INTEGER OPTIMAL SOLUTION FOUND
## <!SOLVER MSG> ----
get_solution(result, x[i, j]) %>%
dplyr::filter(value == 1)
## variable i j value
## 1 x 4 1 1
## 2 x 2 2 1
## 3 x 5 3 1
## 4 x 3 4 1
## 5 x 1 5 1
You can also fix certain indexes.
get_solution(result, x[2, j])
## variable j value
## 1 x 1 0
## 2 x 2 1
## 3 x 3 0
## 4 x 4 0
## 5 x 5 0