You have to use Mathematical Programming to schedule the exams at the Faculty of Science at SDU.

A starting package for the implementation is available in the course git repository $[$GIT$].$ The description of the instances has been moved there in the README.md file of the data$/$ directory.

Let E

be the set of exams for courses that need to have a date scheduled. Let D

be the set of days and R

the set of rooms that are available for scheduling exams. Each exam e

from E

has a set of enrolled students Se

and requires a number of days depending on whether it is written or oral.

Exams that for some reason cannot have overlapping schedules are called conflicting exams. In particular, exams with the same teacher or censor and exams that belong to the same study program are considered conflicting. For an exam i

from E

the set of conflicting exams is denoted Ci

$.$

Exams may share students. The number of students each pair of courses share is denoted by sij

in Z$+0$

for i$,$j

in E

$.$

Further, exams may be joined

$,$ that is$,$ they need to have the same starting date. For an exam i

from E

$,$ the set of exams that must be $$joined$$ with it is denoted Ji

$.$

An exam schedule is an assignment of exams to days and rooms, that is$,$ it is a function $\backslash$sigma :E$->2$D$\backslash$times R

$.$ For example, an exam i

scheduled in days $1$

and $2$

in rooms a

and b

$,$ respectively, can be represented by $\backslash$sigma :i$\{(1,$a$),(2,$b$)\}$

$.$ An exam schedule is feasible if it satisfies the following constraints:

Each exam in input is scheduled to start and finish in the days available.

Each exam is assigned a number of days equal to the number of required days to carry out the exam.

Exams with exam duration more than one day receive consecutive days. No holes due to weekend or holidays are allowed within an exam schedule.Exams that are joined must have schedules starting on the same day. $($The duration for both should be the sum of the duration of the two exams but you can ignore this$).$

Exams that are conflicting must be scheduled in sets of days with empty intersection.

Written exams $($type$==\text{'}$s$\text{'})$ must be scheduled in rooms with enough capacity $($seats$)$ to accommodate all the students registered to the exams $($ideally$,$ there should be a capacity reduction factor to ensure enough distance between students but we will ignore this here, or assume the reduction has already been done$).$

Oral exams $($type$==\text{'}$m$\text{'})$ must be scheduled in rooms with capacity less than $10.$

Optional: Room stability. Exams with duration longer than one day, e$.$g$.$ oral exams, are scheduled all days in the same room.

If a schedule that satisfies all the above constraints is found, then one must maximize the distance between the starting times of exams that share students $($sij$>0$

$)($weekends should be counted in the distance as working days$).$ This optimization task can be achieved in different ways and it is up to you modeling this aspect. It is also desirable finding solutions that address some definition of fairness.

Your Task $($Deterministic Case$)$

Formulate the problem in MILP terms, implement the model and solve the instances of the problems made available for E$23$