next up previous
Next: Mean-Field Annealing Up: No Title Previous: Introduction

The Timetabling Problem

Timetabling is the assignment of time slots to a set of events, subject to constraints on these assignments. The NP-complete professors and classes timetabling problem [7, 12, 13] is a constraint satisfaction problem that can be briefly stated as follows:

For a certain school with tex2html_wrap_inline767 professors, tex2html_wrap_inline769 classes, tex2html_wrap_inline771 classrooms and lecture halls, and tex2html_wrap_inline773 students, it is required to schedule tex2html_wrap_inline775 professor-class pairs within a time limit of tex2html_wrap_inline777 time slots producing a legal schedule. A legal schedule needs to be found such that no professor, class, or student is in more than one place at a time, and no room is expected to accommodate more than one lesson at a time or more students than its capacity.

The constraints for this problem can be hard, medium or soft. The medium and soft constraints have an associated cost (or penalty), and if they are not satisfied, the goal is to minimize this cost. Soft constraints have a lower priority (and thus lower cost) than medium constraints. The hard constraints must be satisfied, so their associated cost must be reduced to zero. A feasible schedule is one that satisfies all the hard constraints.

Hard constraints are usually constraints that physically cannot be violated. This includes events that must not overlap in time, such as:

Another examples are space or room constraints:

Medium constraints are usually considered to those constraints that fall into the gray area between the hard and soft constraints [9]. In our implementation, we define medium constraints to be constraints such as time and space conflicts which, like hard constraints, cannot physically be violated (for example, it is not possible for one person to be in two different classes at the same time). However we consider these constraints to be medium rather than hard if they can be avoided by making adjustments to the specification of the problem. The primary example is student preferences. We cannot expect to be able to satisfy all student class preferences, in some cases, certain students will have to adjust their preferences since certain classes will clash, or will be oversubscribed.

Medium constraints have a high penalty attached to them, although not as high as that associated with the hard constraints. In the final schedule the penalty of these constraints should be minimized and preferably reduced to zero. Some examples of medium constraints are:

Soft constraints are preferences that do not deal with time conflicts, and have a lower penalty (or cost) associated with them. We aim to minimize the cost, but do not expect to be able to reduce it to zero. Some examples are:

Some soft constraints may have higher priority (and thus higher cost) than others. For example, preferences involving teachers will have higher priority than the preferences of students.

The cost function measures the quality of the current schedule and generally involves the weighted sum of penalties associated with different types of constraint violations. The aim of the optimization technique is to minimize the cost function.


next up previous
Next: Mean-Field Annealing Up: No Title Previous: Introduction

Saleh Elmohamed
Tue Apr 29 19:08:49 EDT 1997