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\begin{document}

\bibliographystyle{alpha}
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\begin{flushright}
{\sl SA/TTP}
\end{flushright}

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{\bf 
Simulated Annealing, and non-Euclidean Combinatorial Optimization
Problems: The case of Academic Course Scheduling, Theory and
Implementation.}
\vspace{.25cm}
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\vspace{1cm}

Saleh Elmohamed\\
Geoffrey Fox\\
Paul Coddington\\
Hon Yau\\
Northeast Parallel Architectures Center,\\
Syracuse University\\
Syracuse, NY 13244\\
\{saleh, gcf, paulc, hwyau\}@npac.syr.edu\\
\end{center}

\newpage
\begin{flushright}
{\sl SA/TTP.report}
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{\bf 
Simulated Annealing, and non-Euclidean Combinatorial Optimization
Problems: The case of Academic Course Scheduling, Theory and
Implementation.
}
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Saleh Elmohamed\\
Geoffrey Fox\\
Paul Coddington\\
Hon Yau\\
Northeast Parallel Architectures Center,\\
Syracuse University\\
Syracuse, NY 13244\\
\{saleh, gcf, paulc, hwyau\}@npac.syr.edu\\
\end{center}

\vspace{1.5cm}
\centerline{ABSTRACT}

In this study we tackle a complex problem of academic course 
scheduling (or timetabling) at the university level. The method
used is that of simulated annealing using two different annealing
schedules. A rule-based system is used as a pre-processor in both
approaches, which contributed greatly to the improvement of the final
output. In summary, this multi-phase approach to timetabling produces
much better solutions (schedules) than the single method approach.

\vfill
\noindent{\bf Key words : stochastic updating, annealing, simulated
annealing, cost functions, temperature, annealing schedule, phase
transition, rule-based system.}

\vfill
\centerline{\today}
\vfill

\newpage
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\newpage

\pagestyle{plain}
\pagestyle{myheadings}
\markright{\sl SA/TTP.report}
\section{Introduction}
\setcounter{page}{1}

The primary objective of this study is to derive an approximate solution
to a high-dimensional and non-geometric combinatorial optimization (CO)
problem, namely academic scheduling at the university level. 
 
Our method for tackling the aforementioned problem is simulated
annealing using two different cooling schedules: the standard
geometric cooling, and the geometric re-heating as a function of cost.

--- --- 

Add more here, e.g. write what the following sections
about. 

--- ---

\section{Background and Motivation}

The problem of academic scheduling or timetabling (TTP) has been tackled 
by many researchers \cite{54}, mostly in the field of operations
research. This is a very difficult problem due to its nature as a
multi-constraint satisfaction non-Euclidean combinatorial optimization
problem for which no ``good'' solution has previously been found. 

Lately, there have been a number of {\it non-conventional} (or novel)
methods utilized to tackle this problem from different angles.
Among them are neural networks \cite{32}, simulated annealing
\cite{39, 2}, genetic algorithms \cite{37}, tabu search \cite{38}, and
various combinations of two or more of these techniques. In this
paper, we shall confine ourselves to simulated annealing using the
cooling schedules of geometric annealing and re-heating as a function
of cost. In addition, a rule-based system is used as a pre-processor
in the application of each of these schedules.

--- ---

Give some motivation for these SA solutions as well as for the
preprocessor. 

--- ---

\section{The Timetabling Problem --- description and solution methods}
A problem that does not have a well-defined objective
function (as in the Traveling Salesperson Problem) to optimize and
belongs to the class of constraint satisfaction problems is the
NP-complete {\it teachers and classes} timetabling problem studied
in \cite{2, 3, 27}. Briefly, the problem is stated as follows: 

For a certain school with $N_p$ teachers, $N_q$ classes, $N_x$
classrooms and lecture halls, and $N_s$ students, it is required to 
schedule $N_l$ lessons (defined as teacher-class pairs) within a time
limit of $N_t$ lesson periods (or time slots). 
A legal schedule needs to be found such that no teacher, class,
or student is in more than one place at a time, and no lecture hall is
expected to accommodate more than one lesson at a time or more
students than its capacity.

%% \br{remark.1}
%% From this point on, we will refer to academic scheduling or
%% timetabling as $TTP$ (short for a timetabling system).
%% \er
%% \br{remark.2}
%% Some authors notably Hertz \cite{38} and Eiselt and Laporte \cite{49}
%% refer only to hard constraints as ``constraints'' while the medium and
%% soft are considered as objectives. We don't make such a distinction
%% in this paper.
%% \er

\subsection{The Constraint Satisfaction Problem}

We now briefly describe the general  constraint satisfaction problem
(CSP) which has been widely studied in the Artificial Intelligence
community \cite{9, 10, 11}.

\bd{CSP}\ \
{\it An instance} of a constraint satisfaction problem {\it consists
of}:

\begin{itemize}
\item {\it A finite set of variables, $N_i$, where $i = 1, 2,\ldots,n\;$}.
\item {\it A domain of D of possible values of $N_i$}$\;$.
\item {\it A set of constraints} $\{C(S_1), C(S_2),\ldots, C(S_c)\}\;$.

{\it Each $S_i$ is an ordered subset of the variables, and each
constraint $C(S_i)$ is a set of tuples indicating the mutually
consistent values for the variables in $S_i$}$\;$.

\end{itemize}
\ed

The length of the tuples\footnote{A tuple here is a combination of a
class, teacher, and a classroom. The time periods are assumed to be
given.} in a given constraint will be called the 
{\it arity} of that constraint\footnote{Strictly speaking we also need
to associate an arity with an empty set of tuples, but we will neglect
this special case, in order to simplify the presentation.}. In
particular, unary constraints specify the allowed values for a single
variable, and binary constraints specify the allowed combinations of
values for a pair of variables.
A solution to a constraint satisfaction problem is an assignment
of values to the variables which is consistent with all of the
constraints. Deciding whether or not a given problem instance has a 
solution is NP-complete in general \cite{11}, even when the
constraints are restricted to binary constraints.

\subsection{Constraints of the Timetabling Problem}

In a nutshell, timetabling is the assignment of time slots to a set of
events, subject to constraints on these assignments. Generally,
a constraint places a restriction on when and where classes may be
scheduled. Also as alluded to in \cite{49}, a constraint can be hard,
medium or soft. The medium and soft constraints have an associated
cost and if they are not satisfied, the goal is to minimize this
cost. In regard to the hard constraints, one must derive their
associated cost (or penalty) to zero.

In short, the three types of constraints used are:
\begin{enumerate}
\item Hard - These are constraints that must be satisfied.
\item Medium - These are constraints that should be satisfied, if
possible, but are not required for a valid solution. 
\item Soft - These constraints have a lower priority but always the
preference is to minimize them.
\end{enumerate}

\subsubsection{Hard Constraints}

The most common and important hard constraint is what is sometimes
referred to as an {\it edge} constraint between two vertices or events
(e.g. $e_1$ and $e_2$). This reference is due to the similarity of any
simple timetabling problem with {\it vertex coloring} in graph theory;
which simply states that any given pair of events ($e_1, e_2$) must
not overlap in time. Those events that must not overlap in time
include:

\begin{itemize}
\item two classes taught by the same professor,
\item a class and a recitation or a lab of the same class,
\item classes with students in common.
\end{itemize}

Other general hard constraints include:

\begin{itemize}
\item A class can not be assigned to a particular room unless {\it
size of the room} $\geq$ {\it the class enrollment}. The smaller the
difference between these two sizes, the better the assignment.
\item Daily limits must be specified to restrict professors from
taking more than a specified number of classes per day. Also an
overall limit on how many classes taught by each professor is
enforced.
\end{itemize}

Also, there are other constraints of relevance to students, such as:
\begin{itemize}
\item To sign up for a class or section, it must be open.
\item Eligibility criteria for the section must be met, etc.
\item Do not enroll athletes in courses that conflict with their sport
practice time, etc.
\end{itemize}
These constraints have to be respected at all costs.

\subsubsection{Medium Constraints}

These kinds of constraints fall into, so to speak, the shaded region
between the {\it hard} and {\it soft} constraints in the sense that
they do not directly deal with time conflicts as do the hard type, but
have a high penalty attached to them. Also, this penalty is not as
high as that associated with the hard constraints. In the final
(legal) schedule the penalty of these constraints is minimized and it
is quite preferable to reduce it to $0$.

In short, these kind of constraints should be satisfied but are
nevertheless relaxable. Some examples of medium constraints:

\begin{itemize}
\item For each student, balance {\it Mon, Wed, and Fri} (MWF) schedules as
well as {\it Tue, and Thu} (TuTh)'s.
\item Balance or spread out the lectures over the week.
\item Classes may request contiguous time slots.
\item Classes may be grouped with other classes.
\item The overall schedule should have as few ``holes'' as possible;
that is lessons should be glued together, etc.
\end{itemize}

\subsubsection{Soft Constraints}

Like medium constraints, they also do not deal with time conflicts at
all but unlike the medium constraints, have a low penalty (or cost)
associated with them. Furthermore, this cost is lower than that of the
medium constraints and much lower than that of the hard ones. In
addition, in the final schedule, this cost is minimized but does not
often approach zero. Here are few examples of soft constraints:

\begin{itemize}
\item Balance enrollment in multi-section courses.
\item Lunch and other break times may be specified.
\item Professors may request periods in which their classes are not
taught.
\item Various other preferences of students and (or) professors for
specific rooms even if they are taken from a group or rooms.
\end{itemize}


\subsubsection{More Constraints}

%%%%% *********** May be move the following paragraph somewhere else
%%%%% ***********                    ?????

%% The study case dealt with here is {\it course scheduling} at the
%% university level for both graduate and undergraduate students. Given
%% the various student and course section information, a host of
%% constraints need to be satisfied for any successful scheduling of this
%% nature. In addition to those previously stated in this report, here is
%% a sample ({\it hard and soft}): 
%% ****************************************************************

In addition to those previously stated constraints, we have dealt with
more that are specific to our study case and others are of a generic
nature. A sample follows:

\begin{enumerate}
\item Execlusion constraint: deals with any violation to the
given list of classes that can not be scheduled at the same time or
have any overlapping scheduling time ({\it hard})
\item Section meeting time may not conflict with other ``enrolled''
sections. ({\it hard})
\item An enrollment balance by gender should be obtained in Honors
seminar sections. ({\it soft})
\item Enroll students in ALL parts of a course when associated sections 
(i.e. recitation, labs) are required. ({\it hard})
\item Optimize first choice course enrollments by attempting all first
choice courses for all students before attempting second and third
choices. ({\it soft})
\item Distance/building preference or department-to-building distance.
({\it soft})
\end{enumerate}

Other constraints (hard, medium, and soft) that may need to be taken
into account during the scheduling process are:

\begin{itemize}
\item It is not the case, in general, that all classrooms are
available for all subjects. In reality, many subjects require a
special purpose rooms.
\item Some classes are taught for two or more hours in a row.
\item Group formation: for some subjects the classes are broken up
into option groups temporarily forming new classes.
\item The schedule should have as few ``holes'' as possible; that is
classes should be ``glued'' together.
\item Student with a ``required'' mapping status should occupy a class
seat over a student with other status.
\item Two courses of the same topic may need to be scheduled on two
different working days.
\item Classes may be designated as clashing with other classes because
of common students. 
\item The timetable may consist of an arbitrary number of periods per
day.
\item Some rooms may be declared unavailable during any time period.
\item Class subject combinations may request a preference for certain
periods.
\item Some classes may be grouped with others.
\item In order to schedule a number of classes at the same time, it is
possible to link class tuples together. In this way, the cost of a
move is the sum of the individual costs of each class tuple, however,
they are all moved at the same time.
\end{itemize}

\subsection{Cost Function and the Solution Space}

The cost function measures the infeasibility of the current
solution and may involve the weighted sum of penalties associated with
different types of feasibility violation. Our reference to
``feasibility'' here means that a timetabling schedule is feasible if
it satisfies all of the hard constraints.

Let $C$ be the set of the given constraints. A feasible solution is
defined as a schedule that satisfies $C^{'}$ such that $C^{'}
\subseteq C\;$, provided that all hard constraints are part of
$C^{'}\;$. Also, a neighboring solution is obtained by modifying the
current solution space (i.e. swapping two or more classes in time and
(or) space).
On the question of determining whether a given constraint belongs to
the set, $C^{'}$, or not, roughly we follow these guidelines:

\begin{enumerate}
\item A class should never be assigned to a time period if such an
assignment induces only class schedules that do not satisfy all the
constraints (specifically the hard constraints).
\item A conflict between two or more classes should not be a
sufficient condition for hindering an assignment.
\item
If all courses last one time period, this is equivalent to changing
the color of a vertex, for example.
\end{enumerate}

We consider the following as part of the aforementioned cost function:
\begin{itemize}

\item {\it Distance cost}: proportional to the distance between
where the class is assigned and the building which is housing its home
department or school.

\item {\it Time period cost} consisting of :
\begin{enumerate}
\item class cost, 
\item professor cost, 
\item cost of class spreading over the week and within each of the
weekdays,
\item cost of class glueing and filling in any holes of the timetable,
\item room size cost, and 
\item students cost which is the sum of various costs associated with
each student, such as eligibility criteria and number of classes taken
by the student.
\end{enumerate}

\item {\it Professor cost} is computed from:
\begin{enumerate}
\item cost incurred from  assigning more classes than the maximum
number taught by that professor, plus
\item cost from not having a long enough time interval between courses
taught by the same professor, and
\item cost of assigning to the same professor two or more classes that
overlap in time. 
\end{enumerate}

\item {\it Cost of room size} is computed in a non-linear fashion
depending on the magnitude of the difference between the class
enrollment and the room size.
\end{itemize}

Summing these costs is not strictly necessary, provided that
the components can be combined in some fashion to reflect the quality
of the solution.


\subsection{Complexity of Timetabling}

Timetabling belongs to the class of constraint satisfaction problems
and it is essentially an example of a resource-constrained scheduling
problem, which is an NP-hard problem \cite{12, 25}. 
Here, resources are physical entities, for example: students, rooms,
overhead projectors, and so on. Also time periods can be thought of as
resources.

Traditionally, timetabling has been approached by means of linear
programming (LP) with binary variables plus some heuristics. For
example, if $i$ identifies a teacher, $j$ identifies a time-interval
and $k$ identifies a class then the binary variable $X_{ijk}$ $=$ $1$
if teacher $i$ has class $k$ at time-interval $j$; $X_{ijk}$ $=$ $0$ ,
otherwise.

Using the LP approach, for a set as small as 20 teachers, 30
time-intervals, and 10 classes, the problem would be represented by
over 6000 variables. This makes the problem intractable.
For a much simpler problem, which is the {\it class allocation} (no
timetabling part included), using LP \cite{5, 28}, the
computational load ($n_c$) scales as follows:

\be{complexity}
n_c \propto N_c N_x N_t\;\;,
\ee

where $N_c$ is the number of room categories (i.e. classroom,
auditorium, etc.).

\section{Some Methods of Tackling Timetabling}

\begin{figure}[htb]
\psfig{figure=polytopes.eps,width=432pt,height=228pt}
\caption{An integral polytope has all its vertices at integral points
(points whose coordinates are all integers). It follows that an
integral polytope within the unit hypercube has all its vertices at
$[0-1]$ points.}
\label{poly}
\end{figure}

As we have previously stated that TTP is a multi-constraint
satisfaction problem with hard, medium, and soft constraints each with
their associated cost. A solution from an integer programming point
of view is basically to find any integral vertex within the constraint
polytope $P$, as shown in Figure 
\ref{poly} and Appendix A. To elaborate a bit on this, using the 
work of Schrijver \cite{14, 15}: we think of each constraint as a
hyperplane in the solution space. The set of constraints define a
polytope or (a bounded polyhedron) within which the state vector must
remain if it is not to violate any constraints. The $0$ or $1$
solution points will lie at some of the vertices of this polytope, and
whatever annealing method is used, it must force convergence to one of
these points. The problem with annealing (depending on the schedule
used) is that in this case the best one can expect is convergence to a
vertex, not necessarily a 0 or 1 vertex. 

Polytopes like the one in the permutation matrix mapping for the TSP
are integral, so there is no problem finding valid TSP solutions
(though the minimum tour length is another question altogether). Class
scheduling polytopes are a totally different story: Gee \cite{45} and
some of his colleagues at Cambridge University have done some
experiments on them and found that for timetabling problems as small
as five teachers, five classrooms, five classes, and five time-slots,
the proportion of 0-1 vertices is only about $1\%$. This mean that on
average one has to run a descent procedure about 100 times to find a
valid solution! Some of our experiments with sets of teachers,
classes, classrooms, students and time-slots of various sizes confirm
some of Gee's results and conclusions. 

Timetabling problems can be characterized by their {\it sparseness}.
After all the required lessons have been scheduled, there will be $N_s
= (N_x N_t - N_l)$ spare space-time slots, hence, the sparseness of
the problem is defined as the ratio $N_s / (N_x N_t)$. The sparseness
decreases as  $N_s / (N_x N_t)$ goes to zero or as $N_s$ becomes
smaller and smaller. Furthermore, \cite{45} notes that for sparse
problems, the probability of an arbitrary descent procedure converging
to an integral vertex is near one, though this falls off rapidly for
less sparse problems. For dense problems, there is an additional
correlation involving the problem size: as the problem gets larger
(more classes, teachers, rooms, students, etc.), the proportion of
integral vertices gets smaller. From this we could deduce that a
one-shot descent procedure, operating on an arbitrary ``objective
function'', is not likely to be very successful with difficult
constraint satisfaction problems, most probably converging to a
non-integral vertex of $P$, for example. On the other hand, any
``partial'' or ``imperfect'' solution to a timetabling problem found
by methods such as simulated annealing (using some good adaptive
schedules) is likely to be of some use. Since simulated annealing does
not search over the interior of the unit hypercube, but only over its
vertices, the solution it finds will be a unique timetable, possibly
---as it is the case in our experiments--- violating as few as one or two 
constraints. As part of the post-processing operation, we could at 
least remove some lessons to obtain a valid (or legal) timetable which
fulfills most of the teaching requirements. Furthermore, we could do
few more swaps (or moves) of classes around to arrive at a timetable
meeting all the original requirements.

%% \section{Mapping Timetabling to a Graph Coloring Problem}
%% 
%% \subsection{Graph Coloring}
%% 
%% \bd{graph-coloring}
%% Let $G=(V,E)$ be some graph with $V$ nodes and $E$ edges.
%% A {\it coloring} of $G$ is an assignment of colors to the nodes of $G$
%% such that no two  adjacent  nodes share the same color. More formally,
%% a {\it k-coloring} of $G$ is a mapping $f: V \longrightarrow \{1,
%% 2,\ldots, k\}$ such that $f(u) = f(v)$ only if $(u,v) \not\in E$. The
%% {\it chromatic number} of $G$, denoted $\chi(G)$, is the minimal
%% number of colors necessary to color $G$. An {\it optimal coloring} of
%% $G$ is one which uses exactly $\chi(G)$ colors.
%% \ed
%% 
%% \subsection{The Mapping}
%% 
%% \begin{enumerate}
%% \item The courses are the vertices of a graph.
%% \item The time periods correspond to the colors.
%% \item Two vertices are linked by an edge if the corresponding courses
%% cannot be given at the same time.
%% \item Additional constraints have to be taken into account and mapped
%% onto the graph.
%% \end{enumerate}
%% 

\section{Our Approach in Tackling the Problem}

Given data sets of courses and their days, hours, enrollments, and
instructors; rooms and their capacities, specification, and locations;
distances between buildings; priorities of each building; students and
their courses: the problem is to construct a legal school schedule
satisfying all the hard constraints and minimizing the medium and soft
constraints. Courses that are not assigned to rooms due to conflict in
time or size are left out of the final schedule for the user to
handle manually.

The technique used is a rule-based system with lots of heuristics and
conventional recursion to assist in carrying out course
assignments. The output of the system is partial solution, in a sense
that due to time conflicts not all of the given courses are assigned to
rooms and times slots. In the second phase of operation, this partial
solution along with medium and soft constraints are fed into a second
processing system, which is simulated annealing using two different
cooling methods, to produce the final schedule. 

\subsection{The Preprocessing System}

Here are few examples of some of the implemented rules
dealing with space and hours, type of room, and priority of room:

\subsubsection{Space and Hours}

{\it IF [room(capacity) $>$ course(space-requested)] and [no time
conflict in this room]
THEN
   assign the room to the course:}

[If more than a room having an identical availability time are present
in the database, then choose the one with the minimal magnitude of the
difference between room(capacity) and course(space-requested)].

{\bf\underline{Input:}}
CPS-615 requests space for 75 students and meeting hours of 
[MWF 10:40 - 11:35 am].

In the database we have, for example, the following rooms:

\begin{itemize}
\item room: 1-218 CST, capacity: 45 students, and pre-occupied hours:
none.
\item room: 207 HL, capacity 120 students, and pre-occupied hours: [MW
09:45 - 11:10 am]. 
\item room: Kitt-Aud HBC, capacity: 127 students, and pre-occupied
hours: [TuTh 10:40 - 12:00 am].
\end{itemize}

{\bf\underline{Output:}}
{\it Assign room Kitt-Aud to class CPS-615}. Clearly this assignment
meets the class requirement of size and room availability at the
rquested meeting time.\ \

\subsubsection{Type of Room}

{\it IF [room(capacity) $>$ course(space-requested)] and [no time
conflict in assigning this room] and 
[type(room) matches type(requested)] 

THEN
   assign the room to the course:}

[If identical rooms in type are present in the database, then choose
the one with the minimal magnitude of the difference between
room(capacity) and course(space-requested)].

{\bf\underline{Input:}}
BIO-615 (not biology lab) requests space for 35 students and meeting
hours in [MW 10:40 - 11:35]. 

In the database we have, for example, the following rooms:

\begin{itemize}
\item room: 1-218 CST, capacity: 45 students, type: classroom, and
pre-occupied hours: none.
\item room: 124-SIMS, capacity: 40 students, type: laboratory, and
pre-occupied hours: none.
\end{itemize}

{\bf\underline{Output:}}
{\it Assign room 1-218 CST to class BIO-615}. This assignment meets
the criterion of room type, eventhough size of the second item in the
database is more ideal given the class enrollment. \ \

\subsubsection{Room Priority}

Rooms are classified as either general purpose (GP) which are used by
all, or priority rooms (PR) with a usage based on certain priority of
the class or the department, or special purpose (SP) rooms which are
only assigned to specific users.

Here is a sample of rules showing how these room categories are used:

\begin{itemize}
\item {\it IF [room(priority) == general] and [room(capacity) $>$
course(space-requested)] and [no time conflict in assigning this room]
and [type(room) matches type(requested)]  

THEN  assign any user to this room.}

\item {\it IF [room has a priority user] and [the user is already
assigned] and [room(capacity) $>$ course(space-requested)] and [no
time conflict in assigning this room] and [type(room) matches
type(requested)]

THEN  assign other users.}

\item {\it IF [room(priority) == special purpose] and [user belongs to
the special users of this room] and [room(capacity) $>$
course(space-requested)] and [no time conflict in assigning this room]
and [type(room) matches type(requested)]

THEN  assign the room to the user.}

\item {\it IF [room has a priority user] and [the user is already
assigned] and [room(capacity) $>$ course(space-requested)] and [no
time conflict in assigning this room] and [type(room) matches
type(requested)]

THEN  assign other users.}

\item {\it IF [room(priority) == special purpose] and [user belongs to
the special users of this room] and [room(capacity) $>$
course(space-requested)] and [no time conflict in assigning this room]
and [type(room) matches type(requested)]

THEN  assign the room to the user.}
\end{itemize}

\subsection{Components of the Rule-Based System}

This system consists of a number of heuristics implemented to take
care of various cases that arise when carrying out class
scheduling. We have developed this system specifically for the problem
of academic scheduling. The basic data structures or components of the
system are:

\begin{itemize}
\item {\bf Distance Matrix} of integer values between each
academic department and every other building under use for scheduling.

\item {\bf Course Data Structure} of each course scheduled anywhere in
campus. These structures are capable of linking with each other.
 
\item {\bf Room Data Structure} of each room (regardless of type) involved
in the scheduling process. Like courses, rooms are capable of linking.

\item {\bf Time Periods Data Structures} to keep track of which hour or
time slot is occupied and which is not.

\item {\bf Department Inclusion Data Structure} giving department
inclusion within other larger departments or colleges, etc.
\end{itemize}

\subsection{The Overall Function of the System}

Essentially our rule-based system functions as follows: given data
files of courses, rooms and buildings, department-to-building distance
matrix, and the inclusion data, the system first: builds an internal
data base, second: using this internal database, it carries out the
scheduling process. This process of scheduling involves a number of
essential sub-processes such as checking the distance given in the
distance matrix, checking building, room type, and hours occupied,
checking and camparing time slots for any conflicts, checking rooms
for any space conflict, and keeping track of and updating the hours
already scheduled.

\subsection{The Output of the Rule-Based System}

The first output of the system consists of classes, with their
associated professors and students, assigned to various rooms.
The second output is a list of classes which could not be
assigned due to time and (or) space or conflict(s).
These two outputs are fed to the main method which is simulated
annealing algorithm to yield a final legal schedule.

\subsection{The Rationale for Using a Preprocessor}

It is a good practice when using techniques such as simulated
annealing, to start at a good point in the search space or with a
partial solution of the problem; regardless of the difficulty or
complexity of the problem at hand. Such a starting point can only 
have a positive effect on the overall solution, and that is improving
its quality. Also, a good starting point or a partial solution will
have a positive effect on the convergence of the system, that is,
faster convergence to the final solution.

%% \subsection{Mapping the Problem Components onto Time Periods}
%% 
%% The two sketches in Figure \ref{components-mapping} show roughly how the
%% various main components of the timetabling system are mapped onto the
%% desired time periods.
%% 
%% \begin{figure}[htb]
%% \begin{center}
%% \begin{picture}(380,380)
%% \psfig{figure=tuples-periods.ps,height=350pt,width=370pt}
%% \end{picture}
%% \end{center}
%% \caption{Mapping the problem components onto the time slots.}
%% \label{components-mapping}
%% \end{figure}


\section{Timetabling via Simulated Annealing}

Simulated annealing (SA), see Figure \ref{SA-algorithm}, has been
widely used for tackling different combinatorial optimization
problems, particularly academic scheduling \cite{3, 4, 39}. The results
obtained depend heavily on the cooling schedule used. We initially
used the most commonly known and used schedule, which is the geometric
cooling, but later tried the method of geometric re-heating based on
cost \cite{2}.

A comprehensive discussion of the theoretical and practical details of
SA is given in \cite{17}. It suffices here to say that the elementary
operation in the Metropolis method for a combinatorial problem such as
scheduling is the generation of some new candidate configuration,
which is then automatically accepted if it lowers the cost ($C$), or 
accepted with probability $exp(-\Delta C /T)$, where $T$ is the
temperature, if it would increase the cost by $\Delta (C)$. Thus the
technique is essentially a generalization of the local optimization
strategy, where, at non-zero temperatures, thermal excitations can
facilitate escape from local minima. 

The SA algorithm has some advantages as well as some
disadvantages. Among its advantages are the relative ease of
implementation, the applicability to almost any combinatorial
optimization problem, depending on the cooling schedule(s) used it
usually provides reasonable solutions, and it can easily be combined
with other heuristics, such as rule-based systems, forming quite
useful hybrid methods for tackling a range of complex problems.
One major drawback of the algorithm is the requirement for long run
times.

\subsection{Mapping of Timetabling Into Annealing Algorithm}

The most obvious mapping of TTP into the SA algorithm
involves the following constructs:

Let $M_p$ = maximum number of classes assigned to any professor.
\begin{enumerate}
\item a {\bf state} is a timetable containing the following sets:
\begin{itemize}
\item $P$: a set of professors.
\item $C$: a set of classes.
\item $S$: a set of students.
\item $R$: a set of classrooms.
\item ${\cal I}$: a set of time periods.
\end{itemize}
\item a {\bf cost} or ``energy'' E($P, C, S, R, {\cal I}$) such that:
\begin{itemize}

\item E($P$): is the cost of assigning more than $M_p$ classes to the
same professor, plus scheduling one or more courses that cause a conflict
in the professor's schedule.

\item E($C$): is the cost of scheduling certain classes at/within the
same time period in violation of the exclusion constraint, for
example.

\item E($S$): is the cost of having two or more classes conflict in
time; plus cost of having in the schedule one or more classes that
really don't meet the student's major, class requested, or course
requirements; plus the cost of not having the courses evenly spread
over the week, etc. 

\item E($R$): is the cost resulting from assigning room(s) of the wrong
size and/or type to a certain class.

\item E(${\cal I}$): is the cost of having more or less time periods than
required, plus cost of an imbalanced class assignments (a certain
period will have more classes assigned to than others, etc.).
\end{itemize}

\item A {\bf swap} (or a move) is the exchange of one or more of the
following:

\begin{itemize}
\item Class swapping: class $c_i$ with class $c_j$ in the set $C$ with
respect to time periods ${\cal I}_i$ and ${\cal I}_j$, and/or with
respect to classroom $R_i$ and $R_j$, respectively. 
\end{itemize}

\end{enumerate}
Along with the output of the rule-based system, the algorithm also
takes the following as input:
\begin{itemize}
\item A list of courses that have not been already scheduled and their
associated professors, room categories, etc.
\item A list of rooms provided by the registrar's office.
\item A department to building distance matrix.
\item A list of students and their course preferences.
\item A list of courses that are not allowed to be scheduled
simultaneously.
\end{itemize}

\subsection{More on Simulated Annealing}
SA is a robust technique and has been applied with success to a wide
variety of problems. However, to obtain good results the annealing
runs often require a great deal of computer time. Johnson et
al. \cite{19} criticized much of the earlier research for only 
comparing SA with the simpler local optimization
heuristics (i.e. those using r-opt style moves) and not with the
more competitive Lin-Kernighan (LK) heuristics \cite{48}. They
undertook a systematic comparison of the above three methods'
performance on the graph bisection problem, for low degree random and
random geometric graphs. 
When assessed using equivalent amounts of computer time, SA emerged a
clear winner over simple local optimization for both types of
graph. Against the LK-style algorithm however, SA was only slightly
better for random graphs and worse for the random geometrical class. A
similar investigation was carried  out for the TSP \cite{22, 51}
and the results suggest that, matched for run-time, SA beats 3-opt but
not as good as a well-designed and implemented LK heuristics.

In summary, SA appears competitive with many of the best local
optimization methods. Its major drawback is the requirement for long
run times. Furthermore, the inherently serial nature of the approach
in general precludes any obvious parallel implementation, though
several attempts have been made to overcome this problem such as the
recent work by Frost \cite{16}.

\section{The Annealing Schedules}

Two annealing schedules have been used for our experiments: geometric
cooling and an adaptive re-heating as a function of cost (RFC, for
short). This later schedule is due to Abramson \cite{2} but the basic
idea behind it is actually due to Kirkpatrick et al. \cite{52}.

\subsection{Geometric Annealing}

The new temperature in the SA algorithm in Figure (\ref{SA-algorithm}) is
usually computed according to the following cooling schedule:

\be{cooling}
T^{'}\; =\; \alpha\; T\;,
\ee

where $\alpha$ is typically in the range $0.95 \ldots 0.99$. 
This schedule has two main advantages and that is it is based on a
sound theoretical foundation, and well understood by all researchers
working on annealing. Because of these advantages it is used as a base
line for comparison with other scheduling schemes.

%% 
%% {\bf \underline{Disadvantages}}:
%% 
%% The state of the system is not taken into account in any way, and
%% thus can not adapt the intensity of the search depending on the
%% complexity of the problem.

\subsection{Reheating as a Function of Cost}
The second annealing schedule we use to update the temperature of the
SA algorithm is the method of reheating as a function of cost, RFC for
short. Figure \ref{RFC-schedule} shows all the steps involved in this
schedule.

\subsubsection{Some Background}
\begin{itemize}
\item In combinatorial optimization problems, {\bf phase transitions}
can be observed as sub-parts of the problem are resolved. Ray Lister
\cite{7} , in tackling the TSP, observed that at high temperatures the
gross structure of the solution is resolved, and at low temperatures
the minute details of the solution are resolved.

\item Applying a reheating type procedure, depending on the phase,
would allow the algorithm to spend more time in the low temperature
phases, thus reducing the total amount of time required to solve a
given problem.

\item To calculate the temperature at which a {\bf phase transition}
occurs, it is necessary to compute what is called the {\bf specific
heat} of the substance. A {\bf phase transition} occurs at a
temperature $T_{msh}$ when the specific heat is maximal, and this
triggers the change in the state ordering. 

\item If the best solution found to date is poor (i.e. has a high
energy or cost) then the super-structure may require
re-arrangement. This can be done by raising the temperature to a level
which is higher than the phase transition temperature
($T_{msh}$). Generally, the higher the best cost, the higher the
temperature which is required to escape the local minimum.

\item $T_{msh}$ is the temperature at which the maximum specific heat
occurs, or at which the system changes state.

\item Kirkpatrick et al. \cite{52, 53} made a reference to {\bf
specific heat} which they measured by the variance of solution values
encountered at a given temperature. The higher the variance, the
longer it presumably takes to reach equilibrium, and so the longer one
should spend at the temperature, or alternatively, the slower one
should lower the temperature.

\end{itemize}

\begin{figure}[htb]
\hrule
\hrule
\hrule
\hrule
\hrule
\vspace{.05cm}
\hrule
\vspace{.25cm}
\begin{itemize}
\item Let $C_i$ to denote the cost of configuration $i$, $C(T)$ is the
cost at temperature $T$, and $\sigma(T)$ is the standard deviation at $T$.
\item At temperature $T$, the acceptance probability is: 
\be{T.1}
Pr_i(T) = {e^{-C_i \over {kT}} \over \sum_j{e^{-C_j \over {kT}}}}\,\,.
\ee
\item For a set of configurations ${\cal N}$, the expected cost at
temperature $T$: 
\be{T.2}
<C(T)> = \sum_{i \in {\cal N}} C_i Pr_i(T)\,\,.
\ee
\item The expected square cost at $T$: 
\be{T.3}
<C^{2}(T)> = \sum_{i \in {\cal N}} C_i^2 Pr_i(T)\,\,.
\ee
\item The variance of the cost (or energy) at $T$ is:
\be{T.4}
\sigma^{2}(T) = <C^{2}(T)> - <C(T)>^{2}\,\,.
\ee
\item Using the above derivations, the specific heat: 
\be{T.5}
T_{sh}(i) = {\sigma^{2}(T) \over T^{2}}\,\,, \text{where}\,i \in
\{1,2,\ldots,n\,\}\,.
\ee
\item Now the maximum specific heat:
\be{T.5.1}
T_{msh} = maximum-of(T_{sh}(i))\,\,, \text{where}\,i = 1,2,\ldots,n\,.
\ee
\item The new temperature is set to:
\be{T.6}
T = P \cdot C_b(s) + T_{msh}\,\,.
\ee
\end{itemize}
$P$ is a constant and $C_b(s)$ is the current best cost of the given
configuration $s$. Also, we have used another variant of equation
\ref{T.6} to compute a new temperature and that is, after calculating
$\sigma(T)$ from equation \ref{T.4}, the new $T$ is:
\be{T.6.1}
T = T \cdot e^{-{{0.0001 T} \over \sigma(T)}}\,\,.
\ee
It follows from the Boltzmann distribution (see \cite{41, 13}) that 
\be{T.7}
{d \over {dT}}E_T = {d \over {dT}}C(T) = {\sigma^{2}(T) \over T^{2}}
\ee
which is essentially equivalent to equation \ref{T.5}\,.
\vspace{.25cm}
\hrule
\vspace{.05cm}
\hrule
\hrule
\hrule
\hrule
\hrule
\caption{Reheating as a Function of Cost (RFC) Schedule}
\label{RFC-schedule}
\end{figure}

%%\vspace{0.25cm}
%%\begin{figure}[htb]
%%\centerline{\psfig{figure=Temperature.vs.Tmsp.ps,height=320pt,width=320pt,angle=-90}}
%%\caption{Temperature vs Specific Heat}
%%\label{graph.2}
%%\end{figure}
%%
%%\vspace{0.25cm}
%%\begin{figure}[htb]
%%\centerline{\psfig{figure=Temperature.vs.Acceptance.ps,height=320pt,width=320pt,angle=-90}}
%%\caption{Temperature vs Percentage of Acceptance}
%%\label{graph.3}
%%\end{figure}


\section{Experimental Results}
We view our computation to be of two kinds. One kind is to determine
an acceptable set of parameters that can be used during annealing, and
the second type involving the data sets of our study case. In
addition, the first computations is conducted with respect to the
aforementioned data sets.

\subsection{Schedule-Specific Results}
As mentioned above we have used two cooling schedules for our 
experiments, reheating based on cost and geometric cooling. Each 
schedule takes a set of parameters --- the temperature reduction
constant ($\alpha$), time spent at each temperature, initial
temperature, and for the second schedule a final temperature. Overall,
we found that the effect of the temperature reduction factor was even
less important than was predicted on theoretical grounds. (If it was
replaced with, say, its square root, and the number of moves at the
initial temperature is doubled, the solution quality will be the
same). It was arbitrarily fixed at $0.95$. Since we desire a
comparison of the cooling schedules over a range of run times and
solution qualities, the time spent at each temperature was left as an
independent variable.

For geometric cooling, the initial comparison was with a standard sort 
of stopping criterion \cite{19, 51}: the run was terminated when no
moves had been accepted for three generations <<<***** DEFINE. (other
stopping criteria are considered). Also, we have observed that the
choice of initial temperature poses more of a challenge. To be exact,
the experimental procedure was this: Two data sets were obtained for
both schedules. For the geometric cooling, its set consists of ten
runs made from each of ten starting temperatures anywhere between
$100$ and $500$ with increments of $10$'s. For the reheating schedule,
ten choices of the initial $T$ were used anywhere between $128$ and
$500$ also with increments of $10$'s. 

To summarize, there were two data sets, each having $10$ samples for
each of their $10$ parameter combinations. For each combination the
{\it mean run time} as well as the {\it mean final energy (cost)} over
the $10$ sample runs were computed. If one parameter combination
yielded a longer run time and higher final energy than another, it was
discarded. The remaining combinations were those that were most
efficient, that is, by construction they had the property that as 
time increases, the energy decreases. 

%%% In the output side: the lowest
%%% solution energy actually produced was $2230$ (found in $5$ out of $10$
%%% reheating runs but only once by the geometric cooling).

\subsection{Application-Specific Results}
Our study case was quite large in comparison to the scheduling case
used by Peterson et al. \cite{28}. The largest case they
had dealt with involved around 1000 students, 20 different possible
majors, and an overall periodic schedules (over weeks). In the case of
Abramson et al. \cite{2}, the problems they dealt with were quite small
and they stated that problems involving more than 300 tuples were very
difficult to solve. 

Our study case involved real scheduling data covering three semesters
here at Syracuse University. 
Those of first and second semester were roughly of the same size:
around 13600 students, 1100 professors, staff, and teaching
assistants, 400 rooms (classrooms, seminar rooms and auditoriums), and
2500 classes. With the exception of the number of rooms, the third
semester data was much smaller, around 700 classes  **** FIND NUMBER
of PROFs ****. Along with this, we also have utilized the distance (in
a form of a distance matrix) between various buildings on main
campus. Given the various building preferences of the departments, we
have constructed a matrix $M[I,J]$ between all the academic schools and
departments $[I]$ and all the buildings involved in scheduling
$[J]$. Using this matrix in conjunction with the real ---appropriately
scaled--- distance, a final state matrix is derived and directly used
in the scheduling process.

\subsection{Observations and Analysis}

The main objective here is to generate a solution (i.e. a feasible
timetable) subject to all binding (or hard) constraints and to
minimize the cost incurred form the medium and soft constraints.
In regard to the geometric annealing, when we start from some good
solution rather than from the usual random solution, the initial
temperature must not be too high to avoid reaching equilibrium and
``forgetting'' the supplied initial solution. For totally random
starts, this was found not to be of any help.

We also observed that the efficacy of starting annealing (using either
adaptive or non-adaptive schedules) from a good solution depends not
only on the energy (or cost) of that solution, but also on how it was
obtained, that is, on its structure. For example, \cite{19, 51}
give few thoughts on this subject but as far as we know we have not
come across any kind of analysis on the topic.

Our analysis starts with Figure \ref{graph.1} which shows the
relationship between the annealing temperature and cost when using the
RFC method. The interesting thing to notice is
that the cost begins to decrease rapidly around the temperature of
$30$ and continue decreasing picking up speed approximately at
temperature of $20$. The phenomenon of Figure \ref{graph.1} becomes
more obvious when observing Figure \ref{graph.2}, which clearly show a
maximum specific heat at about temperature $30$. Unlike Figure
\ref{graph.1}, the relationship between temperature and cost in Figure
\ref{graph.3} is sort of typical for geometric cooling. The cost
continues to decrease, albeit slowly, beyond the point shown in the
plot till reaching close to 3000. After that the rate of change
becomes even slower with either taking very short upward moves or
slightly going down but no sharp downward moves, as partly shown
in Figure \ref{graph.4}. This pattern continues for at least 10000
more iterations till termination or reaching a stopping point.

One interesting segment of the overall cost is the cost associated
with rooms which is incurred as a result of the assignment of classes
to rooms. After the initial stage (preprocessing) and before
annealing, a room cost is associated with the given problem
configuration. During annealing, this cost changes in a non-linear
fashion depending on the size of rooms and enrollment of 
classes assigned to those rooms. Figure \ref{graph.5} shows the
relationship between room size and class enrollment, while Figure
\ref{graph.6} shows the relationship between room size and cost. These
three factors (room size, class enrollment and cost) and how they are
related are shown in Figure \ref{graph.7}.

\subsection{Code and Simulations}

The code of the aforementioned two annealing schedules are written in
ANSI C and all the simulations were run using the GNU C compiler on
the IBM RS6000 and the SP2. In regard to the SP2, individual jobs were
run separately on different nodes using different sets of parameters
and if needed also different cooling schedules.

\begin{figure}[htb]
\vspace{0.5cm}
\hrule
\hrule
\hrule
\hrule
\hrule
\vspace{.05cm}
\hrule
\vspace{.25cm}
Let ${\cal N}_{s}$ be the class or neighborhood of solution $s$.
\begin{center}
\begin{enumerate}
\item Generate an initial solution (i.e. schedule) $s$ .
\item Set the initial best solution $s^{*} = s$.
\item Compute cost of $s$ :  $C(s)$.
\item Compute initial temperature $T_0$.
\item Set the temperature $T\; =\; T_0$.
\item While {\em stop criterion} is not satisfied do:
\begin{enumerate}
\item Repeat {\em Markov chain length} (M) times:
\begin{enumerate}
\item Select a random neighbor $s^{'}$ to the current solution, 
$( s^{'} \subset {\cal N}_{s} )\;$.
\item Set $\Delta(C) = C(s^{'}) - C(s)\;$.
\item If ($\Delta(C) \leq 0$ $\{$downhill move$\}$):
\begin{itemize}
\item Set $s = s^{'}\;$.
\item If $C(s) < C(s^{*})$ then set $s^{*} = s$.
\end{itemize}
\item If ($\Delta(C) > 0$ $\{$uphill move$\}$):
\begin{itemize}
\item Choose a random number $r$ uniformly from $[0,1]$.
\item If $r < e^{-\Delta(C) /T}\;$ then set $s = s^{'}$.
\end{itemize}
\end{enumerate}
\item Reduce (or update) temperature $T$.
\end{enumerate}
\item Return the solution  $s^{*}$.
\end{enumerate}
\vspace{.25cm}
\hrule
\vspace{.05cm}
\hrule
\hrule
\hrule
\hrule
\hrule
\end{center}
\caption{The Simulated Annealing Algorithm}
\label{SA-algorithm}
\vspace{0.5cm}
\end{figure}

\vspace{0.5cm}
\begin{figure}[htb]
\centerline{\psfig{figure=non-geometric.Temp.vs.Cost-gnuplot.ps,height=320pt,width=320pt,angle=-90}}
\caption{Cost vs. Temperature for the reheating based on cost schedule}
\label{graph.1}
\end{figure}

\vspace{0.5cm}
\begin{figure}[htb]
\centerline{\psfig{figure=non-geometric.Temp.vs.Tmsh.gnuplot.ps,height=320pt,width=320pt,angle=-90}}
\caption{Specific Heat vs. Temperature for the reheating based on cost schedule}
\label{graph.2}
\end{figure}

\vspace{0.5cm}
\begin{figure}[htb]
\centerline{\psfig{figure=geometric1.Temp.vs.Cost-gnuplot.ps,height=320pt,width=320pt,angle=-90}}
\caption{Cost vs. Temperature for the geometric annealing}
\label{graph.3}
\end{figure}

\vspace{0.5cm}
\begin{figure}[htb]
\centerline{\psfig{figure=geometric.Iterations.vs.Cost-gnuplot.ps,height=320pt,width=320pt,angle=-90}}
\caption{Cost vs. Number of Steps for the geometric annealing}
\label{graph.4}
\end{figure}

\vspace{0.5cm}
\begin{figure}[htb]
\centerline{\psfig{figure=roomSize.vs.classEnroll.gnuplot.ps,height=320pt,width=320pt,angle=-90}}
\caption{Room size vs. class enrollment}
\label{graph.5}
\end{figure}

\vspace{0.5cm}
\begin{figure}[htb]
\centerline{\psfig{figure=roomSize.vs.roomCost.gnuplot.ps,height=320pt,width=320pt,angle=-90}}
\caption{Room size vs. room cost}
\label{graph.6}
\end{figure}

\vspace{0.5cm}
\begin{figure}[htb]
\centerline{\psfig{figure=roomSize.vs.classEnroll.vs.cost.gnuplot.ps,height=320pt,width=320pt,angle=-90}}
\caption{Room size vs. class enrollment vs. room cost}
\label{graph.7}
\end{figure}

%%
%%\begin{table}[htb]
%%\vspace{0.25cm}
%%\hrule
%%\hrule
%%\vspace{0.25cm}
%%The following table illustrates some mean error results of the MFA and
%%Simulated annealing using the RFC schedule, with and without
%%the rule-based system as a pre-processor. Each of these figures is for
%%100 runs of the same size (class, classrooms, students, etc.)
%%timetabling problem.
%%\vspace{0.25cm}
%%$$
%%\begin{tabular}{|l||c|c||c|c||}
%%\hline
%%                 &\multicolumn{2}{|c|}{Mean Field Annealing}&\multicolumn{2}{|c|}{Simulated Annealing} \\ \hline\hline
%%                 & Pre-Processor & No Pre-Processor & Pre-Processor   & No Pre-Processor \\ \hline\hline
%%Mean        &  2.6   &  14.24 &  0.32  & 7.1   \\
%%Percentage  &        &        &        &       \\       
%%Error       &        &        &        &       \\ \hline
%%\end{tabular}
%%$$
%%\caption{Performance figures for MFA and SA in the presence and
%%absence of pre-processor (a rule-based knowledge system).}
%%\label{table.1}
%%\vspace{0.25cm}
%%\hrule
%%\hrule
%%\end{table}
%%\begin{table}[htb]
%%\vspace{0.25cm}
%%$$
%%\begin{tabular}{|l||c|c|c|c||}
%%\hline
%%            & Total Courses & Scheduled & Not Scheduled & \% Scheduled\\ \hline\hline
%%First Term  &     4400      &   2992    &      1408     &     68.00   \\ \hline
%%Second Term &     4082      &   2881    &      1201     &     70.57   \\ \hline
%%Third Term  &     709       &   569     &      140      &     80.25   \\ \hline
%%\end{tabular}
%%$$
%%\caption{Performance figures for three terms using a {\it rule-based
%%system} only. Both class and students constraints are used.}
%%\label{table.2}
%%\vspace{0.25cm}
%%\hrule
%%\hrule
%%\end{table}
%%%%------------------------------- NEW ----------------------------
\begin{table}[htb]
\vspace{0.1cm}
$$
\begin{tabular}{|l||c|c|c|c||}
\hline
                & Total Courses & Scheduled & Not Scheduled & \% Scheduled\\ \hline\hline
First Semester  &     2482      &   1688    &      794      &     68.00   \\ \hline
Second Semester &     2164      &   1527    &      637      &     70.57   \\ \hline
Third Semester  &     709       &   569     &      140      &     80.25   \\ \hline
\end{tabular}
$$
\caption{Averaged figures for three terms using a {\it rule-based
system} only. Both class and students constraints are used.}
\label{table.2.1}
\vspace{0.25cm}
\hrule
\hrule
\end{table}
%%%%-----------------------------------------------------------
\begin{table}[htb]
\vspace{0.1cm}
$$
\begin{tabular}{|l||c|c|c|c||}
\hline
                & Total Courses & Scheduled & Not Scheduled & \% Scheduled\\ \hline\hline
First Semester  &     2482      &   1787    &      695      &     72.00   \\ \hline
Second Semester &     2164      &   1690    &      474      &     78.12   \\ \hline
Third Semester  &     709       &   631     &      78       &     89.00   \\ \hline
\end{tabular}
$$
\caption{Averaged figures using a {\em rule-based system (or expert
system)}. Only class constraints are used.} 
\label{table.3}
\vspace{0.25cm}
\hrule
\hrule
\end{table}

\begin{table}[htb]
\vspace{0.25cm}
$$
\begin{tabular}{|l||c|c|c||}
\hline
                & Scheduled & Not Scheduled & \% Scheduled\\ \hline\hline
SA (geometric)  &   1465    &      1017     &     59.70   \\ \hline
SA (cost-based) &   1644    &      838      &     66.22   \\ \hline
\end{tabular}
$$
\caption{Averaged figures for three algorithms with no
pre-processor used. Both students and class constraints are used and
the data is that of the first term. {\em Note}: Annealing is
carried out on space AND time. Also, all shown figures are averaged
over 10 runs usining first semester data only.}
\label{table.4}
\vspace{0.25cm}
\hrule
\hrule
\end{table}

\begin{table}[htb]
\vspace{0.25cm}
$$
\begin{tabular}{|l||c|c|c||}
\hline
                & Scheduled & Not Scheduled & \% Scheduled\\ \hline\hline
SA (geometric)  &   2407    &      75       &     97.27   \\ \hline
SA (cost-based) &   2444    &      38       &     98.47   \\ \hline
\end{tabular}
$$
\caption{Averaged figures for three algorithms using an expert
system as a pre-processor. Both students and class constraints are
used and the data is that of the first term. {\em Note}:
Annealing is carried out on space AND time. Also, all shown figures
are averaged over 10 runs using first semester data only.}
\label{table.5}
\vspace{0.25cm}
\hrule
\hrule
\end{table}

%%--------------------------------------------------------------------
%% Need to include plots: temperature.vs.cost,
%% professor_cost.vs.timesteps, temperature.vs.delta_cost, 
%% roomsizecost.vs.timesteps, class.1.vs.rmsizecost, class.2.vs.rmsizecost,
%% diff.in.cost_class1-class2(pair of swapped classes),
%% class1_enroll.vs.rmsizecost,  class2_enroll.vs.rmsizecost,
%% class1_capacity.vs.rmsizecost, class2_capacity.vs.rmsizecost
%%

\section{Summary and Conclusion}
This part outlines the basic method and the scheduling constraints.
An academic scheduling problem that is based upon equality constraints 
is considered. 
We have used simulated annealing with two types of cooling schedules
to produce our target timetabling schedules. This target need to be a
valid timetable of classes, professors, and students with no
violations at all of the hard constraints and preferrabaly satisfying
most , if not all, of the soft constraints. The rule-based system
which is used as pre-processor for the simulated annealing
contributed greatly to the improvement of the final outcome. 

We plan to carry out more experiments using perhaps other cooling
schedules and an improved pre-processor system.

\clearpage
%\newpage
%\bibliography{TTP}

\appendix
\renewcommand{\thesection}{Appendix A}
\renewcommand{\theequation}{\mbox{A}\arabic{equation}}
\pagestyle{myheadings}
\markright{Definitions}
\setcounter{equation}{0}
\section{}

From \cite{15}, let $\R^n$ be the space of all n-tuples
$(x_1,\ldots,x_n)$ with real entries $x_i$, and $\R$ is the associated
set of scalars. Likewise for $\R^m$, where $m\neq n$.
\begin{itemize}
\item A subset $P$ of $\R^n$ is a polyhedron if and only if there
exists an ${\it m}$ by ${\it n}$ matrix A and a vector $b \in \R^m$ such
that:  $P = \{ x\in\R^n \mid A x \leq b\}\;\;$. Here, $A x \leq b$
means: $a_1x \leq b_1,\ldots, a_mx \leq b_m$, where $a_1,\ldots, a_m$
are the rows of A. The matrix A may have zero rows, i.e. $m = 0$. In
this case $P = \R^n\;\;$.
\item A subset $P$ of $\R^n$ is called a polytope if $P$ is the convex
hull of a finite number of vectors. That is, there exist vectors
$x_1\ldots\x_t \in \R^n$ such that: $P = conv.hull\{x_1 \ldots x_t\}$.
Also, a subset $P$ of $\R^n$ is a polytope iff it is a bounded
polyhedron. For more information, see references \cite{14, 15}.
\item A polytope $P$ is called integral if each of its vertices is an
integer vector. Clearly, if a polytope $P = \{x | Ax \leq b\}$ is
integral, then the LP-problem $max\{c^Tx| Ax \leq b\}$ has an integer
optimum solution. So in that case,
\be{max-max}
max\{c^{T}x | Ax \leq b ,\;x\;\text{integer} \} = 
max\{c^{T}x | Ax \leq b \}\;.
\ee
\end{itemize}




%% \appendix
%% \newpage
%% \renewcommand{\thesection}{Appendix B}
%% \renewcommand{\theequation}{\mbox{A}\arabic{equation}}
%% \pagestyle{myheadings}
%% \markright{Listing - the source code}
%% \setcounter{equation}{0}
%% \section{}

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