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\begin{document}

\bibliographystyle{alpha}
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%% \begin{flushright}
%% {\sl SA/TTP paper}
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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 C. Fox\\
%% Paul D. Coddington\\
%% Hon W. Yau\\
%% Northeast Parallel Architectures Center,\\
%% Syracuse University\\
%% Syracuse, NY 13244\\
%% \{saleh, gcf, paulc, hwyau\}@npac.syr.edu\\
%% \end{center}

\newpage
\begin{flushright}
{\sl SCCS report - Jan 96}
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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 C. Fox\\
Paul D. Coddington\\
Hon W. 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 cooling
schedules. An expert 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.

--- need to be revised a little ---

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

\vfill
\centerline{\today}
\vfill

\newpage
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\tableofcontents

\newpage
\listoffigures
\listoftables

\newpage
\pagestyle{plain}
\pagestyle{myheadings}
\markright{\sl SCCS report - Jan 96}
\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 this problem is simulated
annealing using two different cooling schedules: the standard
geometric cooling, and the geometric re-heating as a function of cost.

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 particular heuristics or method has been shown to
produce the best (or optimal) results for the problem.

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. None of these
methods are used with as large and complex problem as we have, and
with the exception of \cite{32, 38}, none of the others have used real
world scheduling data. 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. 

In section 2 we give a brief review of TTP and its complexity. Also,
we state the three categories of constraints along with samples of
each that we have implemented in our experiments. Section 3 outlines
an approach to carry out optimization of the problem over integral
polytopes and in section 4 we state our approach to deal with
the timetabling problem. Section 5 is on simulated annealing and the
mapping of the different TTP components onto the annealing cost
function. We state the cooling schedules in section 6 and then discuss
our experimental results in section 7. Lastly, in section 8 we
summarize and conclude the paper.

\section{The Timetabling Problem}
A problem that does not have a well-defined objective
function (as in the traveling salesman problem (TSP), for example) 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 room 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
which has been widely studied in the Artificial Intelligence
community \cite{9, 10, 11}.

An instance of a constraint satisfaction problem consists of:
\begin{itemize}
\item A finite set of variables, $N_i$, where $i = 1, 2,\ldots,n\;$.
\item A domain of D of possible values of $N_i\;$.
\item A set of constraints $\{C(S_1), C(S_2),\ldots, C(S_c)\;$, in
which 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}

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 by Eiselt and Laporte \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 or a legal schedule.
\item Soft - These constraints have a lower priority but always
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 examples of hard constraints are 
\begin{itemize}
\item {\it Room constraint}: essentially says that a class cannot be
assigned to a particular room unless size of the room is greater than
or equal to the class enrollment. The smaller the difference between
these two sizes, the better the assignment. This constraint is
implemented using a non-linear function computing the magnitude of
the difference between the size of room and the class enrollment as
follows: Let $CE$ stands for class enrollment and $RC$ for room
capacity, then $C_{rs}$ (cost associated with the room size) is
computed as follows: 

\be{room-constraint}
C_{rs}=
\twoif{CE - RC} {\text{if} CE > RC\,,}
{\tanh((1.0 - {CE \over RC} - \beta) \gamma \pi) (RC - CE)}
{\text{otherwise.}}
\ee
In our experiments, $\beta$ and $\gamma$ are set to $0.5$ and $0.8$,
respectively.

\item {\it Exclusion 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. Here is how the exclusion cost
$C_e$ is obtained from this constraint: 

$\forall i, j$ such that $i\neq j$, let $c_i$ denotes class(i), $c_j$
denotes class(j), $s(c_i)$ is the starting time of $c_i$,
$end(c_i)$ is the end time of $c_i$, $len(c_i)$ is the duration
of $c_i$, $\epsilon$ is some chosen constant, and $Rlen(c_i,c_j)$ is
the relative length between $c_i$ and $c_j$ as defined below. Also,
the same notation applies to class $c_j$.

\be{exclusion-constraint}
C_{e}=
\fourif {0.0} {\text{if} (s(c_i) > s(c_j))\, and\, (end(c_i) + \epsilon \geq
s(c_j))\,,}
{0.0} {\text{if} (s(c_j) > s(c_i))\,and\,(end(c_j) + \epsilon \geq s(c_i))\,,}
{(end(c_i) - s(c_j)) Rlen(c_i,c_j)} {\text{if} s(c_i) > s(c_j)\,,}
{(end(c_j) - s(c_i)) Rlen(c_i,c_j)} {\text{if} s(c_j) > s(c_i)\,.}
\ee

\be{relative-length}
Rlen(c_i,c_j)=
\twoif {len(c_i) / len(c_j)} {\text{if} len(c_i) \geq
len(c_j)\,,}
{len(c_j) / len(c_i)} {\text{otherwise.}}
\ee

\item Section meeting time may not conflict with other ``enrolled''
sections.
\end{itemize}

Also, there are hard 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 (of course, depending on the sport), and so on.
\item Enroll students in ALL parts of a course when associated sections 
(i.e. recitation, labs) are required.
\end{itemize}
These constraints have to be respected at all costs.

\subsubsection{Medium Constraints}

These kinds of constraints fall into 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, nonetheless,
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 the three-day ({\it Mon, Wed, and Fri}
(MWF)) as well as the two-day ({\it Tue, and Thu} (TuTh)) schedules.
\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 or the separating intervals
are as short as possible.
\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.
\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 Professor or students preferences for specific rooms even if
they are taken from a group of rooms.
\item An enrollment balance by gender should be obtained in Honors
seminar sections.
\item Optimize first choice course enrollments by attempting all first
choice courses for all students before attempting second and third
choices\footnote{This constraint is treated as an optional.}.
\item {\it Distance constraint}: deals with the distance between where the
class is assigned and the building which is housing its home
department. Before actually stating what this constraint is we
will first outline how we have dealt with the aspect of distance. 

Given the various building preferences of the departments, we
have constructed a matrix $M[I,J]$ between all the academic
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. Therefore, unless otherwise indicated, for
a certain department the home building is always the first preference
where the department courses to be assigned.

Let $B$ denotes the set of all buildings and let say $|B| = m$, and
$D$ to denote the set of all departments, $|D| = n$, and $dist$
denotes the distance. Now, for each $d_i \in D$ where $1\leq i \leq n$,
$dist(d_i)$ = vector $\bar{m}<d_i,b_j>$ for ${1 \leq j \leq m}$ of
matrix $M[D,B]$. Also, let $dist(d_i)_{min}$ = minimum of
$\bar{m}<d_i,b_j>$ for ${1 \leq j \leq m}$. Now $C_{d_i}$ (the distance
cost of $d_i$) is computed as follows:

\be{dist-cost}
C_{d_i} = 
\twoif {{dist(d_i)}_{min} {dist(d_i) \over {\sum_{j=1, i\neq j}^m
\bar{m} <d_i,b_j>}} + {dist(d_i)}_{min}} {\text{if} dist(d_i) >
{dist(d_i)}_{min}\,,}
{{dist(d_i)}_{min}} {\text{otherwise.}}
\ee

\item Professor or students preferences for certain
department-to-building distances.
\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 overall given constraints. A feasible solution is
defined as a schedule that satisfies $C^{'}$ such that $C^{'}
\subseteq C\;$, provided that all hard constraints belong to
$C^{'}\;$. Also, a neighboring solution ($s^{'}$) is obtained by
modifying the current solution ($s$) 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 this rule and that is,
a class should never be assigned to a time period if such an
assignment induces only class schedules that do not satisfy the given
constraints (specifically, all the hard constraints). 

The neighborhood of solutions (both feasible and infeasible) that $s$
and $s^{'}$ belong to is denoted by ${\cal N}_{s}$. 

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

\item {\it Distance cost}: obtained from the distance constraint.

\item {\it Time period cost} which consists of the following costs:
cost of class or classes assigned to that period, (for example, the
cost that result from having a certain class and its lab scheduled in
the same time period); cost of class spreading over the week and
within each of the weekdays; cost of class gluing and filling in any
holes of the timetable; and 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.

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

\item {\it Room size cost} is obtained from the room constraint. 
\end{itemize}

Summing these costs is not strictly necessary, provided that the
components (individual costs) can be combined in some fashion to
reflect the overall 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 \cite{12, 25, 58}. Here, resources
are physical entities, for example: students, rooms, overhead
projectors, and so on. Also time periods can be thought of as
resources.

In general, scheduling problems are almost always NP-complete which
means there exist no known polynomially bound algorithm for solving
them optimally. As a result, these problems are often solved by means
of {\it heuristics}: that is solution procedures that focus on finding
a feasible schedule of ``good'' (as opposed to optimal) quality
within an acceptable amount of time.

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. Also, suppose we have a set of data as small as 20
teachers, 30 time-intervals, and 10 classes. If we use the LP approach
to obtain a feasible schedule out of this data then we need to deal
with over 6000 variables which represent the problem. Clearly, as the
size of the input increases so does the number of variables required
to represent it; making the problem intractable.

For a much simpler problem, which is the {\it class allocation} (but
with no timetabling involved, i.e. no allocation of classes to time 
periods), the computational load ($n_c$) from tackling this problem
using the LP method scales as follows \cite{5, 28}: 

\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 AB has all its vertices at integral
points (A = 1.0 and B = 1.0), as shown in the left hand side figure,
but polytope DE in the right hand side figure is not integral since
its ends don't pass through $[0-1]$ 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 or other local
search method is used, it must force convergence to one of these
points. The problem with methods such as annealing (certainly depends
on the cooling 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, for example, 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, Gee \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
violating few soft and/or medium 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.

%% \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; and students
and their courses: the problem was to construct a legal school
schedule satisfying all the hard constraints and minimizing the medium
and soft constraints. Courses that were not assigned to rooms due to
conflict in time or size were left out of the final schedule for the
user to handle manually. We took a multi-phase approach to tackle this
complex problem \cite{49}. In the first phase a feasible solution is
found, and in the second phase the current solution is improved to
obtain a final feasible solution.

In the first phase, the preprocessing phase, we used a rule-based
system giving a feasible, though partial solution as an output. A
partial solution in the sense that due to time conflicts not all of
the given courses were assigned to rooms and times slots. Also, the
preprocessing output was divided into two parts. The first part
consisted of classes, with their associated professors and students,
assigned to various rooms; and the second part was a list of classes
which could not be assigned due to time and/or space conflict. 

In the second phase, the processing phase, the output of the first
phase along with the three types of constraints were fed into the main 
processing method, which was simulated annealing using two different
cooling methods, to produce the final schedule.

\subsection{The Preprocessing System}

As mentioned above, this was an expert (rule-based) system consisting
of a number of rules (or heuristics) and conventional recursion to
assist in carrying out course assignments. We have developed this
system specifically for the problem of academic scheduling. The basic
data structures or components of the system are: first, distance
matrix of integer values between each academic department and every
other building under use for scheduling. Second, course data structure
of each course scheduled anywhere in campus. Also, these structures
are capable of linking with each other. Third, room data structure of each
room (regardless of type) involved in the scheduling process. Like
courses, room structures are also linked with each other. Fourth, data
structures for time periods to keep track of which hour or time slot
was occupied and  which was not. Fifth, department inclusion data
structure giving department inclusion within other larger departments
or colleges.  

The basic function of the system is as follows: given data
files of courses, rooms and buildings, department-to-building distance
matrix, and the inclusion data, using the abovementioned data
structures, the system builds an internal database which in turn is
used in carrying out the scheduling process. This process 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 comparing time slots for any conflicts,
checking rooms for any space conflict, and keeping track of and
updating the hours already scheduled.

\subsection{Motivation and Rationale for Preprocessing}

It is always 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 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 will have a positive effect on the convergence of
the system, that is, faster convergence to the final solution. Our
experimental results, as shown elsewhere in this paper, clearly
support this rationale.

\subsection{Examples}

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
requested 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{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, 57}. 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:
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. Generally, this
step is referred to as class swapping.
\end{enumerate}

Along with all of the necessary constraints and the output of the
expert system, the algorithm also takes as input data the following: a
list of courses that have not been already scheduled and their
associated professors and room types, a list of rooms provided by the
registrar's office, a department to building distance matrix, a list
of students and their course preferences, and a list of courses that
are not allowed to be scheduled simultaneously.

\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-Optimization (or r-Opt) style moves)
and not with the more competitive Lin-Kernighan (LK) heuristics
\cite{48}. They undertook a systematic comparison of the performance
of the above three methods 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{20, 51}
and the results suggest that, matched for run-time, SA beats 3-opt,
also can be quite competitive with LK, but not as good as a
well-designed and implemented LK or the Iterated Lin-Kernighan (ILK)
heuristics \cite{20}.

In summary, depending on the given problem, SA appears competitive
with many of the best local optimization methods. 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}, Boissin and Lutton \cite{55}, and Verhoeven and Aarts
\cite{56}. The most promising approach to parallelizing SA is to
distribute the execution of the various parts of the algorithm over a
number of communicating parallel processors which may contribute to a
significant speedup in the overall runtime of the method. 

\section{The Annealing Schedules}

Two annealing schedules have been used in our experiments to update
the temperature of the SA algorithm in Figure (\ref{SA-algorithm}):
geometric cooling and an adaptive re-heating as a function of
cost. This later schedule was presented in Abramson et al. \cite{2}
but the ideas behind it are actually due to Kirkpatrick et
al. \cite{52, 53} and White \cite{41}.

Let $\alpha$ ($0 < \alpha < 1$) denotes the cooling factor. The first
schedule, geometric cooling, computes the new temperature ($T^{'}$) of
the SA algorithm as follows:

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

Typically the value of $\alpha$ is chosen from the range $0.90 \ldots
0.99$. This schedule has three main advantages and these are: it is
based on a sound theoretical foundation, well understood by all
researchers working on annealing, and the most widely used cooling
schedule. Because of these advantages it is used as a base line
for comparison with other scheduling schemes. Furthermore, our results 
obtained from using it were compared with those of the second
schedule.

%% 
%% {\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.

The second annealing schedule we used to update the temperature of the
SA algorithm is the method of reheating as a function of cost
(RFC). Before stating this schedule we would like to summarize few
relevant points to the concept of specific heat (sh). Specific heat is
not a new concept and in their seminal work, Kirkpatrick et
al. \cite{52, 53} measured it by using 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.

Generally, in combinatorial optimization problems, phase transitions 
can be observed as sub-parts of the problem are resolved. 
In some of the work dealing with the TSP using annealing,
e.g. \cite{7}, the authors often observe that the resolution of the
overall structure of the solution occurs at high temperatures, and at
low temperatures the fine details of the solution are resolved.
As reported in \cite{2}, 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.

In order to calculate the temperature at which a phase transition
occurs, it is necessary to compute the specific heat of the
substance. A phase transition occurs at a temperature $T_{msh}$ when
the specific heat is maximal (msh), and this triggers the change in
the state ordering. 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.
Now, to compute the aforementioned maximum specific heat we employ the
following steps which we refer to as the RFC schedule:

\vspace{0.25cm}
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
$T$.
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
For a set of configurations ${\cal C}$, the expected cost at
temperature $T$: 
\be{T.2}
<C(T)> = \sum_{i \in {\cal C}} C_i Pr_i(T)\,\,.
\ee
The expected square cost at $T$: 
\be{T.3}
<C^{2}(T)> = \sum_{i \in {\cal C}} C_i^2 Pr_i(T)\,\,.
\ee
The variance of the cost (or energy) at $T$ is:
\be{T.4}
\sigma^{2}(T) = <C^{2}(T)> - <C(T)>^{2}\,\,.
\ee
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
Now, the temperature at which the maximum specific heat occurs,
or at which the system changes state is computed as follows:
\be{T.5.1}
T_{msh} = \text{Maximum-of}(T_{sh}(i))\,\,, \text{where}\,i = 1,2,\ldots,n\,.
\ee
The new temperature is set to:
\be{T.6}
T = P \cdot C_b(s) + T_{msh}\,\,.
\ee

$P$ 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 based on not the best cost
found so far but rather on the standard deviation of costs at the
current $T$. A similar approach was taken by Huang et al. \cite{59}
yielding an efficient cooling schedule.

Let $T_j$ denotes the current temperature $T$.  After calculating
$\sigma(T_j)$ from equation \ref{T.4}, the new $T$ which is denoted as
$T_{j+1}$ is as follows: 
\be{T.6.1}
T_{j+1} = T_j \cdot e^{-{{0.0001 T_j} \over \sigma(T_j)}}\,\,.
\ee

Following suggestions by Otten and van Ginneken \cite{60} and Diekmann
et al. \cite{61}, $\sigma(T_j)$ is smoothed out in order to avoid any
dependencies of the temperature decrement on large changes in the
standard deviation $\sigma$. We followed the following the general
form of $\sigma(T_{j+1})$ smoothing:

\be{T.6.0}
\sigma(T_{j+1}) = (1 - \omega) \sigma(T_{j+1}) + \omega \sigma(T_j)
{T_{j+1} \over T_j}
\ee
and set $\omega$ to $0.95$.

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}\,.

\begin{figure}[htb]
\vspace{0.25cm}
\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}
\end{center}
\caption{The Simulated Annealing Algorithm}
\label{SA-algorithm}
\vspace{0.25cm}
\end{figure}

\section{The Strategy in Choice of Moves}
Besides satisfying the constraints stated in this paper, we also have
taken a number of steps throughout the scheduling process to further
eliminate any unnecessary and un-beneficial swaps between various
categories of courses. For example, we made sure that a course of type
lecture cannot be swapped with a course of type laboratory and vice
verse. In a sense, we separated those two types and then
carried out the scheduling of lectures first followed by scheduling of
laboratories, provided that no lecture and its associated laboratory are
scheduled in the same time period. Other productive moves we have
taken is that swapping a higher level course (e.g. graduate) with a
lower level course (e.g. a first or a second year type) is allowed in
general since it results in a very negligible cost. Therefore, the
degree of conflict (or cost) we have dealt with in this case goes from
high to low with the following swaps (highest with the first and
lowest with the last):
\begin{itemize}
\item Swapping two courses of the same year (e.g. two freshman
courses). 
\item Swapping a first year course with one of a second year.
\item Swapping a third year course with one of a fourth year.
\item swapping a fourth year course with a graduate level course.
\item swapping a third year course with a graduate level course.
\item swapping a second year course with a graduate level course.
\item swapping a first year course with a graduate level course.
\end{itemize}

Also, throughout scheduling we have assumed that swapping courses that
belong to different academic departments always involve lower costs
than that of swapping courses belonging to the same department. The
justification for this assumption is that courses that belong to the
same department often tend to have students and instructors in
common. In addition, they usually are scheduled in the same set of
rooms allocated to or chosen by that department. Therefore, part of our
strategy of moves is to limit as much as possible the interdepartmental
course swaps. Also, one of the steps we have taken throughout
scheduling is to schedule larger classrooms and auditoriums first in
order to avoid, if possible, not having large enough rooms later for
those classes with large enrollments (e.g. greater than 35 students). 
One other assumption we have made is that those constraints involving
courses that are not available during certain time period(s) due to the
unavailability of their instructors at those times are harder than
constraints of no overlapping courses involving common student(s). 

Generally, the techniques we have taken here can very well be
considered as prunning the neighborhood or narrowing or limiting the
search space heuristics, contributing to a much improved overall
results.

\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. In some of the runs we did set this independent
variable (or the number of steps at each temperature) to be at least
proportional to the neighborhood size or the size of our data.

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\footnote{A generation is
a full Markov chain at a single temperature, (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. 

\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. 

\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 much 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.3} which shows the
relationship between $\log_2(T)$ and cost for the geometric cooling
and the RFC method. The interesting thing to notice is
that the cost begins to decrease rapidly around $\log_2(T)$ of $3.5$
and continue decreasing picking up speed approximately at
$0$. Figure \ref{graph.2} shows the range of the
maximum specific heat which is between $0$ and $1$. As can be seen,
after some fluctuation the specific heat climbs up above 0.4 and
reaching a maximum just below $0.45$. Unlike the relationship between
$\log_2(T)$ and the RFC cost, we could see that this relationship is
very much typical for the 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. Also,
as shown in Figure \ref{graph.5}, in the first 300 or so iterations
very little reheating was done since the cost is a bit high. Reheating
becomes more frequent as it is shown between steps $400$ and
$800$. As the decrease in cost becomes less and less or no noticeable
decrease at all, the frequency of reheating increases sharply as
observed around the $1100$ iteration step.

Generally, factors contributing to reheating are as follows: The drop
of $T$ below the maximum specific heat combined with a slow, no
decrease, or an increase in cost. To be precise, as $T$ drops more and
more below the maximum specific heat and after a number of specified
iterations, if either cost does not decrease significantly or start to
increase, reheating immediately will occur in an attempt to escape
from that cycle.

Our overall results are shown in tables (\ref{table.1}, \ref{table.2},
\ref{table.3}, and \ref{table.4}). Table \ref{table.1} gives a 
percentage output of the preprocessor (the rule-based system) for a
three-semester set of data as well as the highest and lowest
percentage of scheduled data. We also can see that the size of those
scheduled classes increases as the input (number of classes)
decreases, which was not really surprising since the overall number of
rooms and time slots were kept fixed for the three semesters. In
addition, both class and students constraints were taken into
consideration. Unlike Table \ref{table.1}, Table \ref{table.2} shows
improved results and that was mainly due to not taking students
constraints into consideration and only restricting the system to
class constraints.

In the case of simulated annealing, Table \ref{table.3} shows output
in a percentage form averaged over 10 runs for only the first semester
input and without a preprocessor. Clearly, it shows a rather poor
output which is even lower in quality to that of the rule-based
system. Also, the use of the RCF schedule did not improve the results
by much and remained below those of Table \ref{table.1}. 

In contrast to the first three tables, Table \ref{table.4} shows quite
improved and high quality averaged results for both annealing
schedules. Such excellent results were mainly due to the use of
preprocessed input (output of the rule-based system) to the SA
processor. Also the figures were averaged over 10 runs of annealing
and one out of those runs yielded a perfect schedule, that is, all
classes of the input set were scheduled subject to the given students
and class constraints. Clearly, this tells us that the use of
preprocessing was not only helpful but quite essential when using
methods such as simulated annealing to deal with complex timetabling 
problems. Furthermore, these high quality averaged results state in an 
unambiguous terms that a multi-phase approach, as we explained above,
gives a much better results to TTP then a single-phase approach.

In regard to runtimes for the first semester, the 10-run average of
the preprocessor (the expert system) was over two hours. While it
took an average of 22 hours to carry out the processing using the
geometric cooling schedule and about 16 hours when using the reheating
schedule. For the second semester schedule, it took a 10-run average
of an hour and 50 minutes of preprocessing and an average of almost 20
hours of proceesing using the geometric schedule versus about 15 hours
when reheating was used. These runtimes are clearly much faster than
the 30 days or so time period it usually takes for Syarcuse University to
produce a final schedule (students are not included) for a single
semester. The batch scheduling system used is called `25E' and the
overall scheduling process invloves a number of interactive sub-processes
taking place between the central scheduling office and the various schools
and departments of the university. The purposes of these interactions
are to resolve any conflicts arising from class/room assignments and
to satisfy all categories of constraints in a piecemeal fashion. Also,
when scheduling a certain semester (e.g. autumn 1993), a template of a
previous semester (e.g. autumn 1992) is used as part of the input data,
otherwise, the entire process of course scheduling for a single semester
will certainly take much longer than a month.


\subsection{Code and Simulation Times}

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.

\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.

--- This section needs some workout ---

\section*{Acknowledgments}
The authors are very grateful for the valuable help of Robert
Irwin in converting and formatting the registration data prior to the
scheduling process. We also would like to sincerely thank Karen Bedard
for providing us with the data and answering so many questions we had
about it, and Meg Cortese for providing us with a set of building
constraints and preferences for various departments. Thank you Karen
and Meg.


\vspace{0.25cm}
\begin{figure}[htb]
\centerline{\psfig{figure=Temperature.vs.Specific-heat-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{0.25cm}
\begin{figure}[htb]
\centerline{\psfig{figure=Adaptive_Geometric.TempLog2.vs.Cost-gnuplot.ps,height=320pt,width=320pt,angle=-90}}
\caption{Cost vs. Temperature (log base 2) for the geometric annealing
and the reheating based on cost schedules}
\label{graph.3}
\end{figure}

\vspace{0.25cm}
\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{0.25cm}
\begin{figure}[htb]
\centerline{\psfig{figure=Non-geometric.steps.vs.cost-gnuplot.ps,height=320pt,width=320pt,angle=-90}}
\caption{Cost vs. Number of Steps for the reheating based on cost schedule}
\label{graph.5}
\end{figure}

\begin{table}[htb]
\vspace{0.1cm}
$$
\begin{tabular}{|l||c|c|c|c|c|c||}
\hline
                &  Total  & Scheduled &    Not    & Scheduled & highest   & lowest    \\ 
                & Courses &    No.    & Scheduled & (average) & Scheduled & Scheduled \\
                &         &           &    No.    &    \%     &   \%      &   \%      \\ \hline\hline 
First Semester  &  2482   &   1688    &    794    &   68.00   &   71.75   &   66.95   \\ \hline
Second Semester &  2164   &   1527    &    637    &   70.57   &   72.85   &   68.50   \\ \hline
Third Semester  &  709    &   569     &    140    &   80.25   &   85.35   &   77.95   \\ \hline
\end{tabular}
$$
\caption{Averaged figures for three terms using a {\it rule-based
system} only. Both class and students constraints are used.}
\label{table.1}
\vspace{0.25cm}
\end{table}
\begin{table}[htb]
\vspace{0.1cm}
$$
\begin{tabular}{|l||c|c|c|c|c|c||}
\hline
                &  Total  & Scheduled &    Not    & Scheduled & highest   & lowest    \\ 
                & Courses &    No.    & Scheduled & (average) & Scheduled & Scheduled \\
                &         &           &    No.    &    \%     &   \%      &   \%      \\ \hline\hline 
First Semester  &  2482   &   1787    &    695    &   72.00   &  75.15    &  69.55    \\ \hline
Second Semester &  2164   &   1690    &    474    &   78.12   &  81.10    &  75.00    \\ \hline
Third Semester  &  709    &   631     &    78     &   89.00   &  90.45    &  88.90    \\ \hline
\end{tabular}
$$
\caption{Averaged figures using a {\em rule-based system} ({\em expert 
system}). Only class constraints are used.}
\label{table.2}
\end{table}

\begin{table}[htb]
\vspace{0.25cm}
$$
\begin{tabular}{|l||c|c|c|c|c||}
\hline
                & Scheduled &   Not     &  Scheduled & highest   & lowest    \\
                &    No.    & Scheduled &  (average) & Scheduled & Scheduled \\
                &           &   No.     &     \%     &    \%     &   \%      \\ \hline\hline
SA (geometric)  &   1465    &  1017     &    59.70   &   67.95   &  55.28    \\ \hline
SA (cost-based) &   1644    &  838      &    66.22   &   70.47   &  61.12    \\ \hline
\end{tabular}
$$
\caption{Averaged figures for two 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 using first semester data only.}
\label{table.3}
\end{table}

\begin{table}[htb]
\vspace{0.25cm}
$$
\begin{tabular}{|l||c|c|c|c|c||}
\hline
                & Scheduled &   Not     &  Scheduled & highest   & lowest    \\
                &    No.    & Scheduled &  (average) & Scheduled & Scheduled \\
                &           &   No.     &     \%     &    \%     &   \%      \\ \hline\hline
SA (geometric)  &   2407    &   75      &   97.27    &  98.25    &  94.45    \\ \hline
SA (cost-based) &   2444    &   38      &   98.47    &  100.0    &  95.10    \\ \hline
\end{tabular}
$$
\caption{Averaged figures for two 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.4}
\end{table}

\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}

%% \clearpage
%% \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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\end{thebibliography}
\end{document}

