Maximising Diversity and Balancing Skill

Model introduction

Consider a situation where an instructor wishes to divide students into groups. Each group will be allocated a topic $t$, from a pool of topics $1,\ldots,T$. It is possible that each topic $t$ is repeated a certain number of times across the class. Suppose that the maximum number of repetitions of a topic is $r_{max}$. In total, there are $N$ students in the class.

Suppose that students form their own groups, which they submit through a survey form. In total there are $G$ such self-formed groups; each student appears in exactly 1 group.

In addition, we have the following information about each student:

1. Information that can be used to compute dissimilarities between pairs of students. Examples are:
   • the major of the student (STEM vs. non-STEM),
   • gender,
   • year-of-study, etc.
2. Information on the skill level pertinent to the class, or to the problem they will be working on.

This model allows you to maximise the diversity within a group and minimise the difference in skill within groups.

Objective function

The overall objective function can be written as:

$$\max \quad w_1 \left( \sum_{i=1}^{N-1} \sum_{j=i+1}^N \sum_{t=1}^T \sum_{r=1}^{r_{max}} z_{ijtr} \cdot d_{ij} \right) + w_2 (s_{min} - s_{max})$$

where $w_1$ and $w_2$ are weights. They indicate which half of the objective function should be given priority.

Constraints

Group to topic-repetition combination

First, let us introduce the decision variable of interest:

$$\begin{equation} x_{gtr} = \begin{cases} 1 & \text{if group $g$ is assigned to repetition $r$ of topic $t$} \\ 0 & \text{otherwise} \end{cases} \end{equation}$$

$s_{min}$ and $s_{max}$ are also decision variables. The objective function attempts to minimise the difference between them, ensuring all groups have a similar range of total skill.

This first constraint represents the need for each group to be assigned to exactly one topic-repetition combination:

$$\sum_{t=1}^T \sum_{r=1}^{r_{max}} x_{gtr} = 1, \quad \forall g$$

Defining $z_{ijtr}$

$z_{ijtr}$ is a binary variable, used to pick up whether the pairwise dissimilarity between student $i$ and student $j$ should be included in the objective function calculation.

$$\begin{eqnarray} z_{ijtr} &\le& \sum_{g=1}^G m_{ig} \cdot x_{gtr}, \quad \forall i,j,t,r \\ z_{ijtr} &\le& \sum_{g=1}^G m_{jg} \cdot x_{gtr}, \quad \forall i,j,t,r \\ z_{ijtr} &\ge& \sum_{g=1}^G m_{ig} \cdot x_{gtr} + \sum_{g=1}^G m_{jg} \cdot x_{gtr} - 1, \quad \forall i,j,t,r \end{eqnarray}$$

Number of repetitions per topic

This set of constraints serve to regulate the total number of repetitions for each topic. $r_{min}$ and $r_{max}$ are input variables that the instructor needs to set.

$$\begin{eqnarray} a_{tr} &\ge& x_{gtr},\quad \forall t,r \\ a_{tr} &\le& \sum_{g=1}^G x_{gtr},\quad \forall t,r \\ \sum_{r=1}^{r_{max}} a_{tr} &\ge& r_{min},\quad \forall t \\ \sum_{r=1}^{r_{max}} a_{tr} &\le& r_{max},\quad \forall t \end{eqnarray}$$

Take note that the last constraint is automatically satisfied since {a_{tr}} is binary.

Number of students per group

A similar set of constraints are used to bound the number of students in each eventually assigned group. $n_{min}$ and $n_{max}$ are also input variables; they dictate the bounds on the number of students in each assigned group.

$$\begin{eqnarray} \sum_{i=1}^N \sum_{g=1}^G m_{ig} \cdot x_{gtr} &\ge& a_{tr} \cdot n_{min}, \quad \forall t,r \\ \sum_{i=1}^N \sum_{g=1}^G m_{ig} \cdot x_{gtr} &\le& a_{tr} \cdot n_{max}, \quad \forall t,r \end{eqnarray}$$

Per-group skill levels

We aim to maintain the skill level within each active group using the following constraints. Let $M=\sum_{i=1}^N s_i$.

$$\begin{eqnarray} \sum_{i=1}^N \sum_{g=1}^G m_{ig} \cdot x_{gtr} \cdot s_i + M(1-a_{tr}) &\ge& s_{min}, \quad \forall t,r \\ \sum_{i=1}^N \sum_{g=1}^G m_{ig} \cdot x_{gtr} \cdot s_i &\le& s_{max}, \quad \forall t,r \\ \end{eqnarray}$$

When $a_{tr}=1$, the lower bound applies to the total skill in that active topic-repetition combination. When $a_{tr}=0$, the Big-M term relaxes the lower bound so inactive combinations do not force $s_{min}$ to zero.

Binary and non-negativity constraints

$$\begin{eqnarray} x_{gtr} &\in& \{0,1\} \\ a_{tr} &\in& \{0,1\} \\ s_{min} &\ge& 0 \\ s_{max} &\ge& 0 \end{eqnarray}$$