# In dirkschumacher/rmpk: Model Linear and Quadratic Programs

## Introduction

As a real world example problem we would like to match a group of students to a set of courses with the following constraints:

• Each course has a capacity
• Every student needs to be assigned to exactly one course.
• All students have stated individual preferences on a scale from 1 to 3, where 3 is the most favorable course.

## The data

We have $n$ students:

n <- 40


And $m$ courses with equal capacity. The capacity can vary among courses though.

m <- 4
capacity <- rep.int(11, m) # all have equal capacities


In addition, each student has three preferences. To model this we have a function that gives us three courses for each student. The first component has perference 1, second 2, and third 3:

set.seed(1234)
preference_data <- lapply(seq_len(n), function(x) sample(seq_len(m), 3))
preferences <- function(student) preference_data[[student]]

preferences(1)


The last component we need is a weight functions to make the model formulation easier. This function gives us the preference weighting for a course and student pair.

# the weight of a student choosing a course
# if the course is not among the preferences, the weight is -100000
weight <- function(student, course) {
p <- which(as.numeric(course) == preferences(as.numeric(student)))
as.integer(if (length(p) == 0) {
-100000
} else {
p
})
}


Some examples:

weight(1, 3)
weight(1, 23) # this was not a choice by student 1, so we give it a big penalty


Let's take a look at our random preferences. We plot the number of votes for each course grouped by the preference (1, 2, 3).

library(ggplot2)
library(purrr)
library(dplyr)
plot_data <- expand.grid(
course = seq_len(m),
weight = 1:3
) %>% rowwise() %>%
mutate(count = sum(map_int(seq_len(n), ~weight(.x, course) == weight))) %>%
mutate(course = factor(course), weight = factor(weight))
ggplot(plot_data, aes(x = course, y = count, fill = weight)) +
geom_bar(stat = "identity") +
viridis::scale_fill_viridis(discrete = TRUE) +
geom_hline(yintercept = 11)


## The model

The idea is to introduce a binary variable $x_{i, j}$ that is $1$ if student $i$ is matched to course $j$. As an objective we will try to satisfy preferences according to their weight. So assigning a student to a course with preference 3 gives 3 points and so forth. The model assumes, that the total capacity of the courses is enough for all students.

Here it is in mathematical notation:

$$\begin{equation} \begin{array}{[email protected]{}ll} \text{max} & \displaystyle\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{m}weight_{i,j} \cdot x_{i, j} & &\ \text{subject to}& \displaystyle\sum\limits_{i=1}^{n} x_{i, j} \leq capacity_j, & j=1 ,\ldots, m&\ & \displaystyle\sum\limits_{j=1}^{m} x_{i, j} = 1, & i=1 ,\ldots, n&\ & x_{i,j} \in {0,1}, &i=1 ,\ldots, n, & j=1 ,\ldots, m \end{array} \end{equation}$$

Or directly in R:

library(rmpk)
library(ROI.plugin.glpk)
model <- MIPModel(ROI_solver("glpk"))

# 1 iff student i is assigned to course m
x <- model$add_variable(i = 1:n, j = 1:m, type = "binary") # maximize the preferences model$set_objective(sum_expr(weight(i, j) * x[i, j], i = 1:n, j = 1:m), sense = "max")

# we cannot exceed the capacity of a course
model$add_constraint(sum_expr(x[i, j], i = 1:n) <= capacity[j], j = 1:m) # each student needs to be assigned to one course model$add_constraint(sum_expr(x[i, j], j = 1:m) == 1, i = 1:n)
model


## Solve the model

We will use glpk to solve the above model.

model$optimize()  We solved the problem with an objective value of r model$objective_value().

matching <- model\$get_variable_value(x[i, j]) %>%
filter(value > .9) %>%
select(i, j) %>%
rowwise() %>%
mutate(weight = weight(as.numeric(i), as.numeric(j)),
preferences = paste0(preferences(as.numeric(i)), collapse = ",")) %>% ungroup

head(matching)

matching %>%
group_by(weight) %>%
summarise(count = n())


r nrow(filter(matching, weight == 3)) students got their top preference. r nrow(filter(matching, weight == 2)) students were assigned to their second choice and r nrow(filter(matching, weight == 1)) students got their least preferable course.

The course assignment now looks like this:

plot_data <- matching %>%
mutate(course = factor(j), weight = factor(weight, levels = c(1, 2, 3))) %>%
group_by(course, weight) %>%
summarise(count = n()) %>%
tidyr::complete(weight, fill = list(count = 0))
ggplot(plot_data, aes(x = course, y = count, fill = weight)) +
geom_bar(stat = "identity") +
viridis::scale_fill_viridis(discrete = TRUE) +
geom_hline(yintercept = 11)


## Feedback

Do you have any questions, ideas, comments? Or did you find a mistake? Let's discuss on Github.

dirkschumacher/rmpk documentation built on Jan. 13, 2020, 4:48 a.m.