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\begin{document}

\bibliographystyle{alpha}
\newpage
\begin{flushright}
{\sl SCCS report: Last Draft}  % Get an SCCS number
\end{flushright}
\vspace{.75cm}
\begin{center} 
{\LARGE\bf 
Academic Scheduling using Simulated Annealing
with a Rule-Based Preprocessor
}
\vspace{1cm}

{\large
Saleh Elmohamed\\
Geoffrey Fox\\
Paul Coddington\\}
\vspace{.25cm}
Northeast Parallel Architectures Center\\
Syracuse University\\
Syracuse, NY 13244\\
\vspace{.25cm}
\{saleh, gcf, paulc\}@npac.syr.edu\\
\end{center}

\vspace{1.5cm}
\centerline{ABSTRACT}

In this study we tackle the complex  problem of academic
class scheduling (or timetabling) at the university level. 
We use a multi-phase approach to tackle this problem. In the first
(the preprocessing) phase our heuristics consist of a rule-based
expert system, and in the second (the processing) phase a simulated
annealing algorithm with two different cooling schedules being
implemented and compared. This approach enabled us to obtain valid
schedules for the timetabling problem for a large university using a
complex cost function that includes student preferences. 

\vfill
\noindent{\bf Keywords : scheduling, timetabling, optimization, 
simulated annealing, reheating, cost functions, annealing schedule,
phase transition, expert system.}
% IS there a list of standard words to be used?? no.

%%\vfill
%%\centerline{\today}
%%\vfill

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% Remove these???
%% \tableofcontents
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%% \listoffigures
%% \listoftables

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%% \markright{\sl SCCS 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 
problem, namely the university timetabling problem.

The timetabling problem (TTP) has been tackled by many researchers
\cite{50}, 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 heuristic or method has been shown to produce good results
for large-scale problems. This problem continues to draw interests
from researchers in different quarters: from Leighton's \cite{33}
approach to a large scale scheduling via methods of graph coloring to
de Werra's \cite{21} review of the area to Hertz \cite{29, 38},
Dowsland \cite{3, 4}, Abramson \cite{18} and others applications of
various cases of the problem. A number of different methods and
heuristics have been tried on different instances from high school to
university course scheduling. For small to medium size problems, some
of the methods outlined in \cite{50} have been shown to produce
superior results than others, but as far as we aware of at the present
there are no comprehensive experimental studies comparing directly the
performance of those methods on the same set of a large scale
real-life data. What we have in mind here is an experimental work in
the same caliber of the extensive work carried out by Johnson and
colleagues \cite{19, 20, 42} of simulated annealing. 

The timetabling problem we have studied can be summarized as follows. 
Given data sets of classes and their days, hours, enrollments, and
instructors; rooms and their capacities, types, and locations;
distances between buildings; priorities of each building for different
departments; and students and their class preferences; 
the problem is to construct a feasible class schedule satisfying all
the hard constraints and minimizing the  medium and soft constraints.
Hard constraints are space and time constraints that must be
satisfied, such as scheduling only one class at a time for any 
professor, student, or classroom. 
Medium and soft constraints are student and teacher preferences that 
should be satisfied if possible.

%% Recently 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{27, 28}, simulated annealing
%% \cite{18, 39}, genetic algorithms \cite{37}, tabu search \cite{38, 62}, and
%% various combinations of two or more of these techniques. 
%% None of these methods has been used on as large and complex a problem 
%% as we have addressed in this work, and with the exception of 
%% \cite{38} and current work to schedule all classes at EPFL in
%% Lasussane, Switzerland using an algorithm similar to the tabu search
%% of Costa \cite{62}, none we know of have used a large scale real-life
%% scheduling data.

We used data for the academic year 1993-94 for classes at Syracuse
University. Currently this problem is handled by the university
scheduling department in a semi-automated fashion. A scheduling program 
is used to find a partial solution, and substantial manual effort is 
required to iterate towards a final solution. Also, when scheduling a
certain semester (e.g. autumn 1996), a template of a previous semester
(e.g. autumn 1995) is used as part of the input data.

We tackle this problem using a multi-phase approach. First, a
rule-based expert system is used as a preprocessor to derive a partial
solution. the second phase is the use of simulated annealing using three
different cooling schedules: the standard geometric cooling, adaptive
cooling, and reheating as a function of cost. Having a preprocessor 
in the application of simulated annealing was found to greatly improve
the quality of the solution.  Using this method, we were able to find 
feasible solutions to the complex class scheduling problem for a 
large university. Prior to the approach of this paper, we have used 
mean field annealing of Peterson et al. \cite{26}, an expert system
(rule-based), and simulated annealing applied separately to the same 
problem but observed that none of these methods gave us as good
results as we reported here for the multi-phase approach.

This paper is organized as follows.
In section 2 we give a brief review of the timetabling problem, and 
present the three categories of constraints along with examples of each.
Section 3 outlines some geometric mapping and few methods of tackling
the timetabling problem.
Sections 4 and 5 describe the two methods of rule-based expert system
and simulated annealing we used in our approach dealing with the
timetabling problem. The discussion of the experimental results of our
system appear in section 7, and finally a summary and conclusion are
presented in section 8.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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
\cite{2, 3, 27}. Briefly, the general 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 $N_l$ teacher-class pairs within a time
limit of $N_t$ time slots producing a legal schedule. 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. 
Figure \ref{relationship} sketches the approximate relationship
between the various entities of the timetabling problem.
\vspace{0.70cm}
\begin{figure}[htb]
\centerline{\psfig{figure=scheduling-net.eps,height=200pt,width=280pt}}
\vspace{0.20cm}
\caption{The relationship between various entities of TTP where P:
professors, C: classes, S: students, R: rooms, and I: time
periods. The direction of the arrow indicates the direction of the
assignment.}
\label{relationship}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsection{The Constraint Satisfaction Problem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% We mentioned above that the timetabling problem we have dealt with is
% essentially a constraint satisfaction problem (CSP), so for that we now
% briefly describe the general CSP 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 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}
%
% Assuming the time periods are given, a combination of a class,
% teacher, and a classroom will be referred to as a tuple. Furthermore,
% the length of the tuples in a given constraint will be called the 
% {\it arity} of that constraint. 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. Note that to simplify our
% presentation, we will neglect the case of associating an arity with an
% empty set of tuples. 
% 
% 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. 
A constraint can be hard, medium or soft \cite{49}. The medium and soft
constraints have an associated cost (or penalty), and if they are not 
satisfied, the goal is to minimize this cost. The hard constraints must
be satisfied, so their associated cost must be reduced 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 (and thus lower
cost) but it is preferable to minimize this cost.
\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 events, 
which simply states that a given pair of events ($e_1, e_2$) must not overlap in time. 
The name is due to the similarity of any simple timetabling problem with {\it vertex coloring} 
in graph theory \cite{21, 22}.

Events that must not overlap in time include:

\begin{itemize}
\item classes taught by the same professor,
\item classes held in the same room,
\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}: a class cannot be
assigned to a particular room unless the capacity of the room $R_c$
is greater than or equal to the class enrollment $C_e$. 
The smaller the difference between
these two sizes, the better the assignment. This constraint is
implemented using a non-linear function for the 
cost $C_{rs}$ associated with the room size:  

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

\item {\it Exclusion constraint}: deals with any violation to the
given list of classes that cannot be scheduled at the same time or
have any overlap in the scheduled times. Here is how the exclusion cost
$C_x$ is obtained from this constraint:

$\forall i, j$ such that $i\neq j$, let $c_i$ denote 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_x =
\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
\end{itemize}

Also, there are hard constraints of relevance to students, such as:
\begin{itemize}
\item To sign up for a class it must be open.
\item Eligibility criteria for the class must be met.
\item Enroll students in all parts of a class when associated sections 
(i.e. recitation, labs) are required.
\item Do not enroll athletes in classes that conflict with their sport
practice time (of course, depending on the sport).
\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, but nonetheless
have a high penalty attached to them (though not as
high as that associated with the hard constraints). In the final
schedule the penalty of these constraints should be minimized and 
preferably reduced to zero.

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, Fri}) 
as well as the two-day ({\it Tue, Thu}) 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 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}

Constraints involving classes 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 classes involving
common student(s).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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 classes.
\item Lunch and other break times may be specified.
\item Professors may request periods in which their classes are not taught.
\item Professors may have preferences for specific rooms or groups of
rooms.
\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. Note: we found that when this constraint is
implemented a contradiction will arise with the abovementioned medium
constraint of class spreading over the week.

\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 $M_{IJ}$ between all the academic
departments $I$ and all the buildings involved in scheduling
$J$. Using $M_{IJ}$ in conjunction with the real -- appropriately
scaled -- distance between all buildings
involved in the process , a final distance matrix $Q$ is derived and
directly used in the scheduling process. Also, unless otherwise
stated, for any department its home building is always the first
preference where the department classes 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 $Q$. Also, let $dist(d_i)_{min}$ = minimum of
$\bar{m}<d_i,b_j>$ for ${1 \leq j \leq m}$ of also matrix $Q$. 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
\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \subsection{Cost Function and the Solution Space}
\subsection{Cost Function}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The cost function measures the quality of the current schedule and
generally involves the weighted sum of penalties associated with
different types of constraint violations. Summing these penalties or
costs is not strictly necessary, provided that the components
(individual costs) can be combined in some fashion to reflect the
overall quality of the schedule. 

Let $C$ be the set of the given constraints. A {\it feasible} schedule 
is defined as a schedule that satisfies the set of hard constraints
$C^{'}$, where $C^{'} \subseteq C\;$.
We consider the following as part of the abovementioned cost function:
\begin{itemize}

\item {\it Time period cost} includes the 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;
and cost of class gluing  
% Define??? 
and filling in any holes of the timetable. 

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

\item {\it Student cost} is the sum of various costs associated with
each student, such as eligibility criteria and number of classes taken
by the student.
% Clash of class times???

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

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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, 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, timetabling problems are always NP-complete
 \cite{25}, which means there exists no known polynomially bound
 algorithm for solving them optimally. As a result, these problems are
 often solved by means of {\it heuristics} -- solution procedures that
 focus on finding a feasible schedule of ``good'' (as opposed to
 optimal) quality within an acceptable amount of time.

%--------------
% is the complexity analysis OK?? yes in its brevity but will
% definitely take more than a short subsection to cover all aspects
% of it.

 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. 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 to represent the problem, and 
 $2^{6000}$ possible states. The problem quickly becomes intractable as
 the number of variables increases.

 For the much simpler problem of {\it class allocation}, for which there
 is 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 available.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \section{Some Methods of Tackling Timetabling}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% The timetabling problem we are dealing with can be summarized as follows. 
% Given data sets of classes and their days, hours, enrollments, and
% instructors; rooms and their capacities, types, and locations;
% distances between buildings; 
% priorities of each building for different departments; 
% and students and their class preferences; 
% the problem is to construct a feasible academic 
% class schedule satisfying all the hard constraints and minimizing the 
% medium and soft constraints. 
%
% If the procedure used cannot assign certain classes to rooms and
% timeslots, i.e. cannot find a feasible schedule for all classes, these
% classes are left out of the final schedule and must be assigned
% manually, or else the input variables must be changed. In some cases
% there may be no feasible schedule for the given constraints, and the
% problem must be reformulated to allow a solution. However our goal is to
% provide a feasible schedule of better quality (i.e. lower cost), and in
% less time, than that provided using current techniques.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Previous Work}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Clarify polytope stuff???

% \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}
%
% The TTP is a multi-constraint satisfaction problem with hard, medium, 
% and soft constraints, each with an associated cost. Most of the work out 
% there in this area is to tackle a subset of the problem, for example, 
% examination scheduling or to deal only with scheduling classes without 
% taking into account the issue of student preferences. In addition, we have 
% noticed that very few of the published studies actually used real-life data, 
% and most used randomly generated or artificial sets of data. A number of
% researchers have tackled the small scale but equally complex high
% school scheduling, e.g. \cite{28}. Traditionally, TTP has been tackled
% in the operations research community using methods such as integer 
% programming and network flow \cite{21}. Also studied by reducing it to
% the well-known problem of graph coloring \cite{1, 21, 33}. More recently,
% a number of heuristics and methods such as simulated annealing
% \cite{3, 4}, tabu search \cite{38, 62}, and genetic algorithms
% \cite{37} have been used to tackle various cases of this problem with
% varying degrees of success. 
% 
% 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}. To elaborate a bit on this, citing \cite{14,
% 15, 23}: 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 $N_{sp}
% = (N_x N_t - N_l)$ spare space-time slots, hence, the sparseness of
% the problem is defined as the ratio $N_{sp} / (N_x N_t)$. The sparseness
% decreases as  $N_{sp} / (N_x N_t)$ goes to zero or as $N_{sp}$ 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-time 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, for example we could at least remove some
% classes to obtain a valid (or legal) timetable which fulfills most of
% the teaching requirements. Table \ref{table.02} shows a three-semester
% ratio of sparseness of our problem.
%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Our Approach to the Timetabling Problem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We adopted a multi-phase approach to tackle the the timetabling
problem \cite{50}. In the first phase a partial solution (a feasible
schedule for a subset of the classes) is found using a rule-based
expert system.  In the second phase this partial schedule is improved
to obtain a final feasible schedule, using simulated annealing.

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, regardless of the difficulty or complexity of the
problem at hand. A good starting point generally provides
faster convergence and a better final solution. Our results clearly
support this rationale in this case. Simulated annealing is a very
time-consuming, computationally intensive procedure. Using a
rule-based preprocessor is a way of quickly providing a good starting
point for the annealing process, which reduces the time taken in the
annealing phase, and improves the quality of the result.

To use simulated annealing effectively, it is crucial to use a good 
cooling schedule, and a good method for choosing new trial schedules, 
in order to efficiently sample the search space. We have experimented
with these two areas, which is discussed in the following sections.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The Rule-Based Preprocessor}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In the first phase, the preprocessing phase, we used a rule-based
system giving a partial feasible schedule as an output. A
partial schedule in the sense that due to constraint conflicts not all 
of the given classes were assigned to rooms and times slots. 
The preprocessor 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 conflict. 

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 class 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 values between each academic department and every
other building under use for scheduling. Second, class data structure
of each class 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
classes, 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. Sixth, students structures indicating classes of various
degree of requirements and preferences for each student.

The basic function of the system is as follows: given data
files of classes, rooms and buildings, department-to-building distance
matrix, students data, 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{Example Rules}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Here are few examples of some of the implemented rules
% dealing with space and hours, type of room, and priority of room.
%
% \noindent
% {\bf Space and Hours}
%
% {\it IF [room(capacity) $>$ class(space-requested)] and [no time
% conflict in this room]
%
% THEN assign the room to the class.}
%
% \noindent
% If more than one 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 class(space-requested).
% 
% \noindent
% {\bf Type of Room}
% 
% {\it IF [room(capacity) $>$ class(space-requested)] and [no time
% conflict in assigning this room] and 
% [type(room) matches type(requested)] 
% 
% THEN
%    assign the room to the class.}
%
% \noindent
% 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 class(space-requested).
%
% {\bf 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) $>$
% class(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) $>$ class(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) $>$
% class(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) $>$ class(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) $>$
% class(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}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{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}. 
% Is this previous work discussed???
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 reheating
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 $\Delta (C)$ 
(see Figure~\ref{SA-algorithm}).
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. 

\begin{figure}[htb]
\vspace{0.25cm}
\begin{center}
\begin{enumerate}
\item Generate an initial schedule $s$.
\item Set the initial best schedule $s^{*} = s$.
\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 schedule, 
$( s^{'} \subset {\cal N}_{s} )\;$.
\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 schedule  $s^{*}$.
\end{enumerate}
\end{center}
\caption{The Simulated Annealing Algorithm}
\label{SA-algorithm}
\vspace{0.25cm}
\end{figure}

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.

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} when tackling problems such as the TSP.  

In summary, depending on the given problem, SA appears competitive
with many of the best local optimization methods.

%
% 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. 
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Timetabling Using the Annealing Algorithm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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

\begin{enumerate}
\item a {\bf state} is a timetable containing the following sets:
\begin{itemize}
% Make sure variable names don't clash??? e.g. P, C, I?
\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 $I$: a set of time intervals.
\end{itemize}
\item a {\bf cost} or ``energy'' E($P, C, S, R, I$) such that:
\begin{itemize}

\item E($P$): is the cost of assigning more than maximum number of 
allowed classes $M_p$ to the same professor, 
plus scheduling one or more classes 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 class
requirements; plus the cost of not having the classes evenly spread
out 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($I$): 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 $I_i$ and $I_j$, 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
rule-based preprocessor, 
the simulated annealing algorithm also takes as input data the following: 
a list of classes that have not been already scheduled by the preprocessor
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 class preferences, and a list of classes that
are not allowed to be scheduled simultaneously.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The Annealing Schedules}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Three annealing schedules have been used in our experiments to update
the temperature of the SA algorithm in Figure~\ref{SA-algorithm}:
geometric cooling, adaptive reheating as a function of cost, and
adaptive cooling.
The second schedule was presented in Abramson et al. \cite{2}
but the ideas behind it are due to Kirkpatrick et al. \cite{52, 53}
and White \cite{41}. 

% Other schedules used but not included here is the
% Lundy-Mees schedule \cite{63}.

Let $\alpha$ ($0 < \alpha < 1$) denote the cooling factor. The first
schedule, geometric cooling, computes the new temperature ($T^{'}$) of
the SA algorithm using:
\be{geomtric}
T^{'} = \alpha T\;.
\ee
Typically the value of $\alpha$ is chosen in the range $0.90$ to $0.99$. 
This cooling schedule has the advantage of being well understood, having
a solid theoretical foundation, and being the most widely used annealing
schedule. 
Our results obtained from using this standard cooling schedule will be used
as a baseline for comparison with those using the second method.

%% 
%% {\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) \cite{2}. Before introducing this schedule we first summarize a few
relevant points on the concept of specific heat ($C_H$). Specific heat is
a measure of the variance of the cost (or energy)
values of states 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 
\cite{8} 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
system. A phase transition occurs at a temperature $T(C_H^{max})$ when
the specific heat is maximal ($C_H^{max}$), and this triggers the change in
the state ordering. If the best solution found to date 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 $T(C_H^{max})$.
Generally, the higher the current 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. The steps
involved in this schedule are mostly adapted from the work of van
Laarhoven and Aarts \cite{17} and Abramson et al. \cite{2}.

At each temperature $T$, the annealing algorithm generates a set of 
configurations ${\cal C}(T)$. 
Let $C_i$ denote the cost of configuration $i$, $C(T)$ is the average 
cost at temperature $T$, and $\sigma(T)$ is the standard deviation of the
cost at $T$. 

At temperature $T$, the acceptance probability is: 
\be{T.1}
P_i(T) = {e^{-C_i \over {kT}} \over \sum_j{e^{-C_j \over {kT}}}} \,.
\ee

The average cost is computed as: 
\be{T.2}
<C(T)> = \sum_{i \in {\cal C}} C_i P_i(T) \,.
\ee

Therefore, the average square cost is: 
\be{T.3}
<C^{2}(T)> = \sum_{i \in {\cal C}} C_i^2 P_i(T) \,.
\ee

The variance of the cost is:
\be{T.4}
\sigma^{2}(T) = <C^{2}(T)> - <C(T)>^{2} \,.
\ee

Now, the specific heat is defined as: 
\be{T.5}
C_H(T) = {{\sigma^{2}(T)} \over {T^{2}}} \,.
\ee
The temperature $T(C_H^{max})$ at which the maximum specific heat occurs,
or at which the system undergoes a phase transition, can thus be found.
Reheating sets the new temperature to be:
\be{T.6}
T = K \cdot C_b + T(C_H^{max}) \, ,
\ee
where $K$ is a constant and $C_b$ is the current best cost. 

It is worth pointing out that reheating is done when the temperature 
drops below the phase transition (the point of maximum specific heat)
and there has been no decrease in cost for a specified number of
iterations, i.e. the system gets stuck in a local minimum. Reheating
increases the temperature above the phase transition (see equation \ref{T.6}), 
in order to produce enough of a change in the configuration
to allow it to explore other minima when the temperature is reduced again.  
Also, we would like to point out that the ``smoothing'' approach of
equation \ref{T.6.0} is  similar to an approach used in iterative
methods for sparse linear systems.  Specifically, the parameter
$\omega$ is used to compute the successive over-relaxation  (or SOR)
phase steps in the Gauss-Seidel algorithm where the former is an
extension of the later method \cite{65}.

The third cooling (adaptive) schedule used is to compute a
new temperature based on on the best cost found so far but the
standard deviation of all costs obtained at the current $T$. A similar
approach was taken by Huang et al. \cite{59} yielding an efficient
cooling schedule. Let $T_j$ denote the current 
temperature, at step $j$ of the annealing schedule.
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^{-{{a T_j} \over \bar\sigma(T_j)}} \, ,
\ee

where $a$ is a parameter that we have set to $0.0005$.
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 used the following standard method to provide a smoothed standard
deviation $\bar\sigma$:

\be{T.6.0}
\bar\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 $0.95$.
This smoothing function is used because it follows (from the form of the
Boltzmann distribution, see \cite{41, 13}) that it preserves the key 
relationship:
\be{T.7}
{d \over {dT}}C(T) = {\bar\sigma^{2}(T) \over T^{2}} = C_H
\ee

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The Choice of Moves}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The performance of any application of simulated annealing is highly
dependent on the method used to select a new trial configuration of the
system for the Metropolis update. In order for the annealing algorithm to 
work well, it must be able to effectively sample the parameter space, 
which can only be done with efficient moves.

Let $s$ be the current schedule and $s^{'}$ be a neighboring schedule
obtained from the current neighborhood  space (${\cal N}_{s}$) by
swapping two classes in either time or/and space.

The simplest method for choosing a move is to swap the rooms or
timeslots of two randomly selected classes. However this is extremely
inefficient, since most of the time random swapping of classes will
increase the overall cost, especially if we are already close to
obtaining a valid solution (i.e. at low temperature), and will likely
be rejected in the Metropolis procedure. This low acceptance of the
moves means this simple method is very inefficient, since a lot of
computation is required to compute the change in cost and do the
Metropolis step, only to reject the move.

What is needed is a strategy for choosing moves that are more likely to be
accepted. A simple example is in the choice of room. If we randomly choose
a new room from the list of all rooms, it will most likely
be rejected, since it may be too small for the class, or an auditorium 
when, for example, a laboratory is needed. One possibility is to
create a subset of all the rooms which fulfill the hard constraints on
the room for that particular class, such as the size and type of
room. Now we just make a random selection for a room for that class
only from this subset of feasible rooms, with an acceptance
probability that is sure to be much higher. In addition, each class
in our data set comes with a ``type-of-space-needed'' tag which is
used along with other information to assign the class to the right room.
This effectively separates the updates into independent sets based on
room type, so for example, laboratories are scheduled separately from
lectures. In our method we carry out the scheduling of lectures first,
followed by scheduling of laboratories making sure that during the
course of this process no lecture and its associated laboratory are
scheduled in the same time period.

In effect, we have embedded a simple expert system into the annealing
algorithm in order to improve the choice of moves, as well as using a more 
complex expert system as a preprocessor for the annealing step.

To improve further on the move strategy, we can take the subset of
possible move choices that we have created for each class, and choose
from them probabilistically rather than randomly. There may be certain
kinds of moves that are more likely to be effective, so our move
strategy is to select these moves with a higher probability.  
For example, swapping a higher level class (e.g. graduate) with a
lower level class (e.g. a first or a second year type) generally has a 
higher acceptance, since there is little overlap between students taking
these classes. Furthermore, we have experimented with two kinds of
swaps, those that only involve classes offered by the same department
or college and the second, swaps between classes of different departments
and colleges. As illustrated in Figure \ref{class-swaps}, swapping is 
done as follows: Given a certain college or a school, let say
school(i), we randomly pick a department, let say A. Based on the
given constraints, we carry class swapping within department A making
sure that no cycling occurs. The second step is to swap classes
between A and another randomly selected department, let say C, within
school(i). The third step of class swapping takes place within
department C. This process of swapping continues within the same
school provided all room and time constraints are satisfied and also
no cycling of the same departments or classes occurs. We refer to
this phase of swapping as ``local'' since it only involve classes
either of the same department or classes of departments that belong to
the same school or college. As previously mentioned, associated with
each department a list of buildings where it assign its classes to. The
first building in the list is the home building and always the first
preference for class assignment. If the tag name ``others'' appears in
the list then the department has an option of swapping its classes
with those of any other department of not only the same school but
with others as well. This ``global'' swapping has to be done in such a
way that all constraints involved and room preferences for both
classes are satisfied. 

The plots of Figures \ref{graph.6} and \ref{graph.6.1} clearly show
that the overall rate of acceptance of swapping classes of the same 
departments or schools is higher than the overall rate of acceptance 
swapping classes of different departments of different schools. 
We could see that one of the reasons for this is that
classes of the same departments or schools are associated with the same
subset of rooms that satisfies those constraints involving distance and
department preference. Therefore, the likelihood of satisfying the
constraints of room availability, capacity, and type within that group
of rooms is much greater than if it is the otherwise case. In addition,
departments and schools in our test data always prefer to have their
classes (especially the lower and the upper level) spread out into as
many time slots as needed, in turn minimizing, if not totally
eliminating, any overlap that may occur otherwise. Consequently, the
likelihood of any time conflict occurring after any department
two-class swap is quite low. Also, as shown in Figures \ref{graph.6}
and \ref{graph.6.1}, those acceptance rates drop off with the decrease
in temperature. 

Another move strategy we have followed is that since constraints
involving capacity of rooms are hard ones, we schedule larger classes
first to try to avoid not having large enough rooms later for those
class sections with large enrollments. 

Generally, the swap methods we have taken here can be considered as 
heuristics for pruning the neighborhood or narrowing the search space,
which provides much more efficient moves and in turn an overall
improvement in the results. 

\vspace{0.25cm}
\begin{figure}[htb]
\centerline{\psfig{figure=class-swaps.eps,height=330pt,width=330pt}}
\vspace{0.20cm}
\caption{A pictorial view of the class swaps. Three kind of swaps are
shown: first, class swapping within the same department; second, class
swapping between two departments; three, class swapping between two
departments of different schools. The first two types of swaps are
classified as (local) within the same dept/school while the last one
(global) involving different schools.}
\label{class-swaps}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Experimental Results}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Our computations were done with a number of goals in mind. The main
objective was to provide a schedule which satisfied all hard
constraints and minimized the cost of medium and soft constraints, 
using real-life data sets for a large university. 
We also aimed to find an acceptable set of annealing parameters 
and move strategies for general timetabling problems of this kind,
and to study the effect of using a preprocessor to provide the annealing
program with a good starting point. Finally, we wanted to make a
comparison of the performance of the two different cooling schedules,
geometric cooling and reheating based on cost.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The Timetabling Data Set}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Our study case involved real scheduling data covering three semesters
at Syracuse University. The size and type of the three-semester data
is shown in Table \ref{table.01}. Nine types of rooms were used and
they are: auditoriums, classrooms, computer clusters, conference
rooms, seminar rooms, studios, laboratories, theaters, and
un-specifies types. Staff and teaching assistants are considered part
of the set of professors. In addition, third semester (summer) data
was much smaller, however, there were additional space/time
constraints and fewer number of available rooms. Our data was quite
large in comparison to data used by other researchers, for example,
\cite{18, 27, 28}. Peterson and colleagues high school data \cite{26,
27, 28} consists of approximately 1000 students, 20 different possible
majors, and an overall periodic school schedule (over weeks). In the
case of Abramson et al. \cite{18}, their data set was created randomly
and relatively small. Also they stated that problems involving more
than 300 tuples were very difficult to solve. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Experimental Runs}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We have observed that the choice of initial
temperature poses more of a challenge. To be exact, the experimental
procedure was this: For the geometric cooling as well as the reheating
schedule, $10$ simulation runs were made from each of the following
starting temperatures: 50, 100, 150, 200, 250, 300,
350, 400, 450, 475, 485, 490, 495, 500, 505, 510, 515, 520, 525, 550, 600, 650, 700,
850, 900, 1000. The challenge stems mainly from the fact that in our simulations
we apply a preprocessor to give us a good starting point in the search
space. So using either very small or very large initial temperature
can disturb this desirable starting state leading to inferior final
solution or a solution with high cost. 
As shown in Figure \ref{graph.6.2}, the smaller values of the set
yield more inferior solutions than the higher values, and generally 
final cost is the lowest around the intermediate values of the set. In
particular, the temperature of $500$ gives the lowest final
cost. Consequently Figures \ref{graph.3}, \ref{graph.2},
\ref{graph.5}, \ref{graph.6}, and \ref{graph.6.1} are all based on an
initial temperature of $500$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The Cooling Schedule - Some Analysis}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We have used three cooling schedules for our 
experiments, reheating based on cost and geometric cooling. Each 
schedule takes a set of parameters --- the initial temperature, the
temperature reduction constant ($\alpha$), and time (number of iterations)
spent at each temperature. We also needed to specify a criterion for 
stopping the procedure when it was believed that the solution was
good enough that we do not expect to obtain much improvement by
extending the run.

The temperature reduction factor $\alpha$ and the the number of moves at 
each temperature $N_T$ are related, since we expect that if $\alpha$ is 
replaced with its square root (so the number of temperatures in the
cooling schedule is doubled), and $N_T$ is halved, the solution quality is 
expected to be about the same, as long as $\alpha$ is large enough.
Usually $\alpha$ is chosen to be in the region $0 \le \alpha < 1$. 
We found only a small variation in the results for this parameter range,
and decided to arbitrarily fix $\alpha$ to be $0.95$, and use $N_T$ as
a variable with which to compare solution quality.  
In some of the runs we set this variable to be at least proportional to 
the neighborhood size, or the size of our data, and in general we observed 
that this yielded slightly better results.

In general, specifying an initial temperature for annealing is usually
a straightforward procedure. Infinite temperature means that the
changes are always accepted (i.e. the acceptance ratio is 1), which
produces random configurations. In practice the initial temperature
should not be too large, or else the cooling will take too long, and
the annealing will spend too much time at higher temperatures where
little useful work is being done. So the initial temperature is
usually chosen to be the lowest possible value that still gives a high
acceptance ratio (greater than $0.5\%$, for example). 

Choosing an initial temperature is much more difficult when we use
a preprocessor to provide a good starting configuration
rather than the usual random initial configuration. In that case,
we must be very careful that the initial temperature and the
acceptance ratio are not so high that the good initial configuration
is randomized to the point where is loses its usefulness. However the
initial temperature must be high enough so that we can still make
effective changes in the configuration. In our abovementioned chosen
sets of experimental starting temperatures we have observed (see
Figure \ref{graph.6.2}) that when using the 2 intermediate
temperatures, 500 and 650, and particularly 500 for both schedules,
the system simulation comes as close as satisfying those criteria,
more so than when other temperatures of the set are used. 

For geometric cooling as well as for the adaptive schedule, the
initial comparison was with a standard type of stopping criterion
\cite{19, 51}: the run was terminated when no moves had been accepted
for five generations (a generation is a full Markov chain at a single
temperature). 

% \subsection{The Choice of Moves} ???
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Observations and Analysis}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Johnson et al. \cite{51} noted in their SA implementation for the TSP
that the number of steps at each temperature (or the size of the Markov
chain) needed to be at least proportional to the ``neighborhood'' size
if they were to obtain worthwhile tour quality. From our experiments
we found that the above hypothesis raised in \cite{51} to also hold
eventhough the nature of our problem is different than
TSP. Furthermore, in few tests for one semester we fixed number of
classes and professors but varied number of rooms and time slots and
found that the final result improved as the size of Markov chain becomes
more proportional to a combination of number of classes, number of
rooms and number of time slots. In other words, as the ratio 
of Markov chain to the combined number of classes, rooms, and
timeslots approaches 1.0 a more improved final result is obtained. We
also observed the same behavior when we fixed the number of rooms and
time slots but varied number of classes. In regard to the relation
between the initial temperature and number of iterations per
temperature, in few runs we observed that as the number of iterations
per temperature decreases it is more preferable to start with a lower
initial temperature, and this is more so when using the geometric
cooling schedule than adaptive annealing. On this last point, from our
experiments involving the preprocessing phase, in general we have not
observed such a relationship.

Also, we 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}
gave few thoughts and hypotheses on this subject as mentioned above
but as far as we know we have not come across any kind of analysis on
the topic.

% This is not an analysis either!!
% How does it depend on structure?? How can you tell??
% THIS DOESN'T EXPLAIN ANYTHING!!
% How does specific heat fit in?? 

Our analysis starts with Figure \ref{graph.3} which shows the
relationship between $\log_2(T)$ and cost for both the geometric
cooling and the adaptive method. The change in cost for the geometric
schedule seems to be more ``smoother'' or uniform and less steeper
than that of the adaptive schedule. Furthermore, for the adaptive
schedule after reaching a certain point around $\log_2(T)$ of $4$ it
stabilizes and the rate of change in cost becomes almost unnoticable. 

Table \ref{table.01} lists all major components of the data and Table
\ref{table.02} shows the rates of sparseness of the three-semester
data which greatly depends on number of time slots and rooms used. The
level of sparseness increases as the overall size of the data,
particularly, number of rooms, time slots, and number of classes.

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. So the
system does quite well for small sets of classes even when 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 three semesters 
without a preprocessor. Clearly, it shows a rather poor output which
is even lower in quality to that of the rule-based system of Table
\ref{table.1}. Also, the use of the adaptive (cost-based) schedule did
not improve the results by much and remained below those of Table
\ref{table.1}. Furthermore, the adaptive annealing gave an overall 
better average result than that of the geometric schedule for the
three semester.

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 for the cost-based schedule $6$ out of those runs for the third
semester set yielded a perfect schedule, that is, all classes of the 
input set were scheduled satisfying the given class and students 
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, one as explained above, gives a much better
results to TTP then a single-phase approach. 


\subsection{Code and Simulation Times}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The annealing programs were written in ANSI C, and all the simulations 
were run on a number of IBM RS/6000 workstations, and on a 12-processor 
IBM SP2, which uses RS/6000 processors.  % specify CPU???
For the SP2, individual jobs were run separately on different nodes
using different sets of parameters, or different cooling schedules, or
just different random numbers. 
The average time for running the preprocessor (the expert system) for
the first and second semester data was about two hours, and the average
time for each simulated annealing run was approximately 20 hours for
the geometric cooling schedule and about 14 hours using the reheating
schedule. 

These runtimes are much faster than
the 30 days or so time period it usually takes for Syracuse University to
produce a final schedule (not including student preferences) for a single
semester, using a semi-automated procedure which involves substantial 
manual effort to iterate to a solution. 

\section{Conclusions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We have successfully applied simulated annealing to the difficult
problem of academic scheduling for a large university. Feasible
schedules were obtained for real data sets without requiring 
enormous computational effort.

We found that we could only obtain successful results by using 
a preprocessor to provide a good initial schedule to the annealing
program. We used a fairly complex rule-based system for the 
preprocessor, however the type of preprocessor may not be crucial.
Other fast heuristics could possibly be used, for example a graph
coloring approach, or it may be possible to just utilize the schedule
from the same semester for the previous year.

Another critical part of the annealing program was the choice of moves.
We used techniques that drew from our work on the rule-based system in
order to carefully select good moves that have a relatively high
probability of acceptance, rather than wasting computational effort on
randomly chosen moves, most of which would be rejected. 
We tested two types of cooling schedules: the standard geometric
cooling, and a cost-based annealing method. This former method
outperformed the former for the larger sets but didn't do as well
smaller one.

\section{Future Work}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In order to avoid dealing with a rather complex expert system, we
will try to use graph coloring heuristics as worked out in \cite{1}
as a preprocessor. Also, as a preprocessor, we will investigate the
possible use of previous year's templates. For the processor, we would
like to try other methods, particularly tabu search, genetic
algorithms, and other methods outlined in Johnson et
al. \cite{51}. Furthermore, using the same set of data we plan to
carry an intensive comparative study on these and perhaps other
methods exploring the various characteristics and constraints of the
timetabling problem. We are investigating these possibilities.

\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 would like to thank Seong-Joon Yoo for the useful
discussion on the pre-processing phase and Hon Yau for  the  useful 
comments and suggestions contributing to the improvement of the presentation 
of the contents. Also, we sincerely thank Karen Bedard for providing us with the 
data and answering so many questions we had about it, Meg Cortese for providing us 
with a set of building constraints for various departments, and Ben Ware, Vice President for
Research and Computing at Syracuse University, for his support and 
encouragement.

%%%%%%%%%%%%
% TABLES
%%%%%%%%%%%%

\newpage

\begin{table}[htb]
\vspace{0.1cm}
$$
\begin{tabular}{|l||c|c|c||}
\hline
                  & First Semester & Second Semester & Third Semester \\ \hline\hline
Rooms             &     509        &      509        &     120        \\ \hline
Classes            &     3839       &      3590       &     687        \\ \hline
Professors        &     1190       &      1200       &     334        \\ \hline
Students          &     13653      &      13653      &     2600       \\ \hline
Buildings         &     43         &      43         &     11         \\ \hline
Schools/Colleges  &     20         &      21         &     17         \\ \hline
Departments/      &     143        &      141        &     108        \\
Course Prefix     &                &                 &                \\ \hline
\end{tabular}
$$
\caption{Size of a three-semester data.}
\label{table.01}
\vspace{0.25cm}
\end{table}

\begin{table}[htb]
\vspace{0.1cm}
$$
\begin{tabular}{|l||c||}
\hline
Academic               & sparseness \\ 
Time Period           &    ratio   \\
                               &            \\ \hline\hline 
First Semester       &    0.50    \\ \hline
Second Semester &    0.53    \\ \hline
Third Semester      &    0.62    \\ \hline
\end{tabular}
$$
\caption{A three-semester sparseness ratio each computed as
$N_{sp}/(N_x N_t)$ is shown. The smaller the ratio, the less sparse
the problem is. Refer to section (3) for more details. Here, the ratio
is based on $N_t$ = 15 time slots each with a duration of 55 minutes. The
first slot starts at 8:00 am and the last slot ends at 9:45 pm.}
\label{table.02}
\vspace{0.25cm}
\end{table}

\begin{table}[htb]
\vspace{0.1cm}
$$
\begin{tabular}{|l||c|c|c|c||}
\hline
Academic        &  Total     & Scheduled  & highest       & lowest    \\ 
Time Period     & Classes & (average) & Scheduled & Scheduled \\
                         &               &    \%         &   \%             &   \%      \\ \hline\hline 
First Semester  &  3839    &   69.20      &   74.90       &   66.60   \\ \hline
Second Semester &  3590   &   70.85   &   76.15   &   68.55   \\ \hline
Third Semester  &  687    &   90.35   &   93.77   &   86.45   \\ \hline
\end{tabular}
$$
\caption{Percentage of scheduled classes, averaged over 10 runs,
for three terms using a {\it rule-based system} only. 
Both class and student constraints are used.}
\label{table.1}
\vspace{0.25cm}
\end{table}

\begin{table}[htb]
\vspace{0.1cm}
$$
\begin{tabular}{|l||c|c|c|c||}
\hline
Academic        &  Total  & Scheduled & highest   & lowest    \\ 
Time Period     & Classes & (average) & Scheduled & Scheduled \\
                &         &    \%     &   \%      &   \%      \\ \hline\hline 
First Semester  &  3839   &   73.19   &   76.65   &   70.23   \\ \hline
Second Semester &  3590   &   76.00   &   79.00   &   72.80   \\ \hline
Third Semester  &  687    &   93.70   &   96.80   &   91.00   \\ \hline
\end{tabular}
$$
\caption{Percentage of scheduled classes, averaged over 10 runs,
for three terms using a {\it rule-based system} only. 
Only class constraints are used.}
\label{table.2}
\end{table}

\begin{table}[htb]
\vspace{0.25cm}
$$
\begin{tabular}{|l|l||c|c|c||}
\hline
Academic        &   Algorithm     & Scheduled  & highest   & lowest    \\
Time Period     &                 & (average)  & Scheduled & Scheduled \\
                &                 &    \%      &    \%     &    \%     \\ \hline\hline
First Semester  & SA (geometric)  &   60.55    &   65.50   &   52.80   \\
                & SA (cost-based) &   66.85    &   70.40   &   61.30   \\ \hline
Second Semester & SA (geometric)  &   61.65    &   66.72   &   56.00   \\ 
                & SA (cost-based) &   67.84    &   71.68   &   63.91   \\ \hline
Third Semester  & SA (geometric)  &   84.12    &   88.50   &   68.00   \\
                & SA (cost-based) &   89.14    &   92.87   &   70.25   \\ \hline
\end{tabular}
$$
\caption{Percentage of scheduled classes, averaged over 10 runs of the
same initial temperature and other parameters, for three terms using
{\it simulated anealing} only, with no preprocessor and a random
initial configuration. Both class and student constraints are used.}
\label{table.3}
\end{table}

\begin{table}[htb]
\vspace{0.25cm}
$$
\begin{tabular}{|l|l||c|c|c||}
\hline
Academic        &   Algorithm     & Scheduled & highest   & lowest    \\
Time Period     &                 & (average) & Scheduled & Scheduled \\
                         &                 &    \%     &    \%     &    \%     \\ \hline\hline
First Semester  & SA (geometric)  &   91.90   &   95.92   &   80.00   \\
                          & SA (cost-based) &   98.17   &   98.97  &   90.66   \\ \hline
Second Semester & SA (geometric)  &   92.85   &   95.95   &   80.40   \\
                              & SA (cost-based) &   98.90   &   99.70   &   91.40   \\ \hline
Third Semester  & SA (geometric)  &   96.51   &   99.89   &   88.90   \\
                           & SA (cost-based) &   100.0   &   100.0   &   100.0   \\ \hline
\end{tabular}
$$
\caption{Percentage of scheduled classes, averaged over 10 runs of the
same initial temperature and other parameters, for three terms using 
{\it simulated anealing with a rule-based preprocessor.} Both class
and student constraints are used.} 
\label{table.4}
\end{table}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% FIGURES
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newpage
\vspace{0.25cm}
\begin{figure}[htb]
\centerline{\psfig{figure=adaptive-and-geometric.TempLog2.vs.Cost.ps,height=280pt,width=280pt,angle=-90}}
\caption{Cost vs. temperature (log to the base of 2) for the geometric and
adaptive (RFC) schedules.}
\label{graph.3}
\end{figure}

\vspace{0.25cm}
\begin{figure}[htb]
\centerline{\psfig{figure=Temp_log2.vs.Cost-after-cooling.ps,height=280pt,width=280pt,angle=-90}}
\caption{Cost after cooling vs. temperature (log base 2) for the reheating based on cost
schedule.}
\label{graph.2}
\end{figure}

%%\vspace{0.25cm}
%%\begin{figure}[htb]
%%\centerline{\psfig{figure=geomtric.Cost.vs.Steps.ps,height=280pt,width=280pt,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-geom.geom.Steps.vs.Cost.ps,height=280pt,width=280pt,angle=-90}}
\caption{Cost vs. number of steps for the adaptive (RFC) and the
geometric schedules. As shown the range of steps is relatively small
in comparison to the Markov chain which is about a factor 20 as large
as the intermediate value of the range.}
\label{graph.5}
\end{figure}

\vspace{0.25cm}
\begin{figure}[htb]
\centerline{\psfig{figure=Adaptive.log_T.vs.same-diff-avg.ps,height=280pt,width=280pt,angle=-90}}
\caption{Temperature vs. rate of acceptance when classes of
different schools/depts and the same schools/depts are swapped. This is
using the adaptive (RFC) schedule. As shown, the figure includes the
average rate as well.} 
\label{graph.6}
\end{figure}

\vspace{0.25cm}
\begin{figure}[htb]
\centerline{\psfig{figure=Geometric.log_T.vs.same-diff-avg.ps,height=280pt,width=280pt,angle=-90}}
\vspace{1.0cm}
\caption{Temperature vs. rate of acceptance when classes of different
and same schools or depts are swapped. This is using the geometric
schedule. As shown, the figure includes the average rate as well.} 
\label{graph.6.1}
\end{figure}

\vspace{0.25cm}
\begin{figure}[htb]
\centerline{\psfig{figure=Log-initial_T.vs.Final_cost.ps,height=280pt,width=280pt,angle=-90}}
\vspace{1.0cm}
\caption{Initial temperature vs. final cost for both schedules: the geometric
annealing and the adaptive (RFC). Each of the final costs is an
average of a 10-run at the given initial temperature. Furthermore, the
category of class swaps in these runs is either local or global or both.}
\label{graph.6.2}
\end{figure}

\vspace{0.25cm}
\begin{figure}[htb]
\centerline{\psfig{figure=K.vs.Final-Cost.ps,height=280pt,width=280pt,angle=-90}}
\vspace{1.0cm}
\caption{Final cost vs. the value of ``K'' in the reheating equation. K is multiplied by the best
cost found so far.}
\label{graph.6.3}
\end{figure}

\vspace{0.25cm}
\begin{figure}[htb]
\centerline{\psfig{figure=a.vs.Final-Cost.ps,height=280pt,width=280pt,angle=-90}}
\vspace{1.0cm}
\caption{Final cost vs. the value of the parameter (a) in adaptive cooling schedule.}
\label{graph.6.4}
\end{figure}

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\end{document}

