Simple Timetabling Problem.
Simplified model for timetabling problems consisting of a set of
resources, a set of activities, and a set of dependencies between the
activities. Time is divided into time slots with the same duration.
Every slot may have assigned a constraint, either hard or soft: a hard
constraint indicates that the slot is forbidden for any activity, a
soft constraint indicates that the slot is not preferred. We call
these constraints �time preferences�. Every activity and every
resource may have assigned a set of time preferences, which indicate
forbidden and not preferred time slots.
Activity (which can be, for instance, directly mapped to a lecture) is
identified by its name. Every activity is described by its duration
(expressed as a number of time slots), by time preferences, and by a
set of resources. This set of resources determines which resources are
required by the activity. To model alternative as well as required
resources, we divide the set of resources into several subsets �
resource groups. Each group is either conjunctive or disjunctive: the
conjunctive group of resources means that the activity needs all the
resources from the group, the disjunctive group means that the
activity needs exactly one of the resources (we can choose from
several alternatives). An example can be a lecture, which will take
place in one of the possible classrooms and it will be taught for all
of the selected classes. Note that we do not need to model conjunctive
groups explicitly because we can use a set of disjunctive groups
containing exactly one resource instead (the set of required resources
can be described in a conjunctive normal form). However, usage of both
conjunctive and disjunctive groups simplifies modelling for the users.
Resource is also identified by its name and it is fully described by
time preferences. There is a hard condition that only one activity can
use the resource at the same time. For instance, such resource can
represent a teacher, a class, a classroom, or another special resource
at the lecture timetabling problem.
Finally, we need a mechanism for defining and handling direct
dependencies between the activities. It seems sufficient to use binary
dependencies only that define relationship between two activities. We
defined three temporal constraints: the activity finishes before
another activity, the activity finishes exactly at the time when the
second activity starts, and two activities run concurrently (they have
the same start time).
The solution of the problem defined by the above model is a timetable
where every scheduled activity has assigned its start time and a set
of reserved resources that are needed for its execution (the activity
is allocated to respective slots of the reserved resources). This
timetable must satisfy all the hard constraints, namely:
every scheduled activity has all the required resources reserved,
i.e., all resources from the conjunctive groups and one resource
from each disjunctive group of resources,
two scheduled activities cannot use the same resource at the same
no activity is scheduled into a time slot where the activity or
some of its reserved resources has a hard constraint in the time
all dependencies between the scheduled activities must be
Furthermore, we want to minimize the number of violated soft
constraints in the time preferences of resources and activities. We do
not formally express this objective function; these preferences will
be used as a guide during the search for the solution.