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A HYBRID-BASED FRAMEWORK FOR CONSTRAINT SATISFACTION OPTIMIZATION PROBLEMS1 Wee-Kit Ho Andrew Lim Department of Computer Science National University of Singapore Republic of Singapore Abstract Scheduling and timetabling are commonly faced problems in most businesses and organizations. Both of these problems fall under the domain of constraint satisfaction optimization problems (CSOP), which involves finding a solution that satisfies all hard constraints, while at the same time maximizing some weighted sum of the soft constraints. Current constraint satisfaction techniques fare poorly in terms of soft constraints satisfied, while optimization techniques cannot ensure the feasibility of the final solution. In this paper, we propose a framework for CSOP that combines both constraint satisfaction and optimization techniques into a hybrid algorithm, called the combined method (CM). We test our framework on an exam-timetabling problem (ETTP) using actual data. Our results show that CM can be expected to produce better results than using a single technique alone.

INTRODUCTION From delivery schedules to duty rosters, seminar timetables to security details, the timetabling problem is encountered by all manner of businesses and corporations. The task of creating a timetable can be seen as a combination of two computational problems: the constraint satisfaction problem and the optimization problem. The constraint satisfaction problem involves the satisfaction of all the hard constraints, which is the criterion that determines if a given timetable is feasible. The optimization problem involves the satisfaction of soft constraints, which offers a measure of the quality of the produced timetable. Therefore, the timetabling problem can be viewed as a constraint satisfaction optimization problem (CSOP). Several approaches have been proposed to solve this problem. These methods generally focus on either the constraint satisfaction or the optimization aspect of the problem. As a result, several methods that ensure the feasibility of the solution give poor results when its quality of solution is judged. Conversely, techniques that give good solutions in terms of soft constraints may produce infeasible solutions. This paper proposes a hybrid-based framework for solving the constraint satisfaction optimization problem that makes use of a combination of constraint satisfaction and optimization algorithms. This combined method (CM) first uses search heuristics (a technique often employed for optimization problems) to find a high-quality solution that is not necessarily feasible. It then employs a selection algorithm with consistency checking (a constraint satisfaction problem technique) to convert the solution to one that is feasible while retaining much of the quality of the original solution. We demonstrate the effectiveness of this framework by studying its use on an examination-scheduling problem based on real data from the National University of Singapore. The next section gives an overview of the constraint satisfaction optimization problem, along with the commonly used techniques to solve the problem. We then introduce our hybrid-based combined method framework. The details of the experiments performed using this framework are then given and our experimental results are analyzed. Finally, we present our conclusions.

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Keywords: 2001 — Twenty-Second International Conference on Information Systems

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Ho and Lim/A Hybrid-Based Framework

THE CONSTRAINT SATISFACTION OPTIMIZATION PROBLEM The constraint satisfaction optimization problem (Tsang 1993), as the name implies, is a combination of two types of problems. The first is the constraint satisfaction problem (CSP). A CSP consists of: • • •

A set of variables X = {x 1, …,xn} For each variable xi, a finite set Di of possible values (its domain) A set of hard constraints restricting the values that sets of variables may take simultaneously

A solution to the CSP is an assignment of values to all variables such that every constraint is satisfied. Timetabling and scheduling problems, which are NP-Hard (Garey and Johnson 1979), may be modeled as a CSP. For example, the creation of a weekly nurse duty roster can be viewed as a CSP by treating each of the nurses’ duty periods as the set of variables and the week’s timeslots as the domain for each task. The hard constraints would prevent the same nurse from taking two duties in the same timeslot, restrict certain specialized jobs to only the trained nurses, and so on. Only if all of the hard constraints are satisfied is the solution considered feasible. In real-life scenarios, however, there is usually another set of soft constraints. These are constraints that may be violated while still retaining the feasibility of the solution. However, the quality of two feasible solutions may be measured by a weighted sum of its soft constraint violations. Taking the nurse duty roster analogy further, soft constraints may include not having more than three consecutive duty periods, spreading the number of public holiday duties evenly among all nurses, and so on. The best fulfillment of these soft constraints, as measured by a weighted sum, is an optimization problem.

Consistency Techniques Several methods have been devised to tackle constraint satisfaction problems. The generate-and-test paradigm (GT) systematically generates possible value assignments for each variable and then tests if it satisfies all constraints. More efficient and common is the backtracking paradigm (BT), which attempts to extend partial solutions toward a complete solution, re-assigning previous values when conflicts occur. The common disadvantage of both GT and BT is the late detection of conflicts, which may result in much redundant work done during the search. Therefore, today’s CSP algorithms generally incorporate a consistency technique. Consistency algorithms make use of some degree of forward search to determine if a given value assignment violates a constraint. The earliest work on consistency algorithms (Mackworth 1977) described the NC-1, AC-1, AC-2, and AC-3 algorithms. Node consistency (NC-1) pre-processes every node ni corresponding to a variable vi in the constraint graph to ensure that for every value in the current domain of vi, each unary constraint on vi is satisfied. Arc consistency (AC-1, AC-2, and AC-3) pre-processes all arcs in the constraint graph to ensure that for every arc (vi, vj), for every value in the current domain of vi, there is some assignment to vj that does not violate any binary constraints between vi and vj. AC-1 is inefficient, as it will revise all arcs when any one arc is revised, while AC-2 appears to be more complicated. AC-3 has a neater solution, in which only those arcs that are affected by the revision are revised again. As a result, the AC-3 algorithm has long been a mainstay in the field of constraint satisfaction problems. By integrating systematic search algorithms such as GT and BT with consistency techniques, it is possible to get more efficient constraint satisfaction algorithms. For instance, one approach makes use of a variable ordering method (VOM) that ranks each task to be scheduled by some weighted criteria and places them in a priority queue. The highest ranked task in the queue is successively inserted into the schedule, and consistency checking is performed. This method has proven to be effective in timetabling and scheduling applications.

Search Heuristics Several search heuristics have been devised to address NP-Hard optimization problems. Generic search heuristics have proven popular in several areas, including scheduling and timetabling applications, because they are easily adaptable to a wide range of problems. Examples include simulated annealing (SA), genetic algorithm (GA), tabu search and more recently the Ant Colony system. These heuristics attempt to maximize (or minimize) a particular quality metric, generally expressed in the form of a weighted sum, by successively altering solutions while taking into account the entire global landscape. This is distinct from “greedy” algorithms that generally choose the best values from the current search position. Such greedy algorithms tend to find solutions that are the best of the local neighborhood of solutions, but may not be the best overall. 2

2001 — Twenty-Second International Conference on Information Systems


Much research has been done on the use of generic algorithms for timetabling and scheduling problems. However, there are problems with this approach. Due to the global viewpoint of generic algorithms, it is generally difficult to make even minor changes to a generated solution while retaining its desirable qualities. Furthermore, since these algorithms attempt to maximize the value of a metric, there is no convenient way to ensure that any particular constraint is always satisfied. Also, there is no assurance that a feasible solution will be found should one exist. Other search heuristics have been employed in the field of scheduling and timetabling algorithms. One of the more commonly employed heuristics models timetabling problems is a graph coloring problem (GCP). The objective of the GCP is to color the nodes of a given graph in k colors such that no two adjacent nodes have the same color. Scheduling problems in their simplest form can be viewed as a GCP, where each node represents a task, each color represents a timeslot, and each edge (vi, vj) indicates that vi and vj should not be placed within the same timeslot. The GCP is a well-researched problem, and several efficient graph coloring heuristics have been devised. Consequently, such graph coloring methods (GCM) have been employed for scheduling and timetabling problems with some success.

THE COMBINED METHOD Obviously, the different approaches to solving the CSOP have their own advantages and disadvantages. While constraint satisfaction algorithms with consistency techniques are able to effectively generate feasible solutions, their quality in terms of soft constraint satisfaction is likely to be poor. Generic search heuristics do a good job of creating solutions that maximize some quality metric, but suffer when the several hard constraints must be satisfied. We now propose a hybrid-based framework for the constraint satisfaction optimization problem, which we call the combined method (CM). This model makes use of both a generic search heuristic as well as a constraint satisfaction algorithm. In this way, we hope to retain the desirable qualities of both approaches to obtain a solution that is superior to using either approach alone. Figure 1 shows the combined method framework.

Step 1:

Step 2 (Optioal):

Step 3:

Search Heuristic e.g., GA, GCM

Post-Optimizer e.g., Tabu Search

Selection Algorithm e.g., VOM

+

Consistency Algorithm e.g., AC-3

A Feasible Timetable Figure 1. Combined Method Framework The combined method involves three steps. In the first step, we make use of a search heuristic to find a solution that maximizes the desired metric. We make use of a post-optimizer in the optional second step to further tune and improve the solution found in the first step by the search heuristic. At this point, the solution found may have violated some hard constraints, making the solution infeasible. In step 3, we invoke a selection algorithm combined with a consistency algorithm to alter the solution into a feasible one with minimal changes. In this way, we retain much of the quality of the initial solution. 2001 — Twenty-Second International Conference on Information Systems

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CM IN ACTION: EXAM TIMETABLE SCHEDULING The exam timetabling problem (ETTP) is a typical scheduling problem. The task involves scheduling a number of examinations within the set of venue-session tuples. Each examination has a candidature, and each venue has a capacity. The two hard constraints are: 1. 2.

Two exams with common candidates may not be scheduled in the same session. The total number of students taking all the exams in a venue-session tuple must not exceed the capacity of the venue.

While the fulfillment of the above two constraints determine the feasibility of the solution, there are several possible soft constraints. We demonstrate the effectiveness of the combined method on the ETTP, making use of student registration data for both academic years 1997/1998 and 1998/1999 for the National University of Singapore (NUS).

Step 1: Genetic Algorithm and Graph Coloring Method In the first step of our framework, a search heuristic is used to find a solution that attempts to maximize our quality metric. We examine two such heuristics. The first is the genetic algorithm (GA), an adaptive heuristic search method based on population genetics (Marin 1998). GA consists of a population of chromosomes that evolve over a number of generations and are subjected to genetic operations at each generation (Algorithm 1). 1.

Generation ← 0

2.

Generate an initial population of size M and set to current population P

3.

While GA has not converged or terminated a. b. c. d.

e. 4.

Generation ← Generation + 1. ChildCount ← 0. Initialize new population NP to empty. While ChildCount < Size(P) i. Randomly select 2 parents, S1 and S2 from P with fitter parents having a higher probability of being selected. ii. Obtain child solutions by performing Genetic Operations on S1 and S2. iii. Evaluate the child solutions. iv. Insert the unique child solutions to NP. v. ChildCount ← ChildCount + No. of child solutions generated P ← NP

Terminate GA Algorithm 1: Genetic Algorithm

GA starts off with a set of initial solutions of size M. In order to search other points in the search space, variation is introduced into the population chromosome by using genetic operators. These operators search the problem space exploring and exploiting information present in the chromosome population by selecting and recombining primarily offspring that have high fitness value. Mutation is used to insert new variations to the new population. This process iterates for many generations until one of the termination criteria is met. Generally, the termination criteria of a GA are: • •

Convergence within a given tolerance, or Realization of the maximum number of generations to be simulated.

Traditional, simple GA operators such as crossover and mutation are commonly used to construct new solutions from pieces of old ones in such a way that, for many problems, the population steadily improves. Crossover is a simple operation that first selects a crossover point randomly and then swaps corresponding parts after the crossover point of the two parents. Mutation randomly alters a chromosome position with a very low probability. This modification injects variability to the population and slows down the converging process of the population. 4



In this method, we can directly represent each chromosome (a timetable solution) as a string of values. A particular position of the string identifies a corresponding examination and the value at that position denotes the timeslot to which this examination is assigned to. However, using this representation, we cannot ensure the feasibility of the solution when we use traditional GA operators. This is because when the crossover operation is performed, there might be more exams being assigned to a given timeslot, thereby violating the venue capacity constraints. Furthermore, if two particular exams are to be held together as specified, performing a crossover may destroy this property possessed by the original chromosomes. The second method is the graph coloring method (GCM). Here, we use the Iterative Greedy Heuristic (Fu et al. 2000) to find a good coloring solution. 1.

Let G be the problem graph. Vertices are ordered based on the vertex ordering. Let c be the number of examinations. Let n be initialized to 0.

2.

Perform a coloring on G using a greedy heuristic. Let c' be the number of colors used in this solution. If c' < c then i. c ← c' ii. n←0

3.

Group the vertices by the coloring found in step 2. Reorder the vertices in G according to their groups such that all vertices in the same group are consecutively ordered. The order of the group is determined based on some group ordering.

4.

Increment n. If n < LIMIT, go to Step 2. Algorithm 2. Graph Coloring Method

The upper bound to the number of sessions required is the number of exams to be scheduled and this is set as the initial value of c, which keeps track of the least number of colors used so far. n counts the number of non-improving iterations, and is initialized to 0. We arrange the initial ordering of the vertices in descending order based on the following measure:

Wt (v ) = where E' is the set of edges connected to v.

∑ w(e )

e∈E '

Next, we iteratively create a color and greedily assign exams to this color until no exams can be assigned to this color. If the result of greedy coloring gives fewer colors than the previous best, we then update the current best value c, and reset the counter n to 0. In step 3, we reorder the colors simply by reversing the order of the colors while keeping those vertices with the same color together in the sequence. Reordering vertices in this way and then recoloring them based on the new sequence is likely to produce a better coloring solution and can never produce a coloring requiring more colors (Hertz and Werra 1991). The value of LIMIT determines the number of non-improving iterations that are made before the algorithm terminates. Besides being a simple and efficient algorithm, the main advantage of graph coloring is that it can effectively minimize the number of sessions used for our ETTP. However, the use of this algorithm is based on the assumption that we are dealing with only the two hard constraints. We are not able to incorporate other types of examination constraints into this method.

Step 2: Tabu Search Step 2 is a post-optimization step. For this study, we made use of a standard tabu search to make swaps of all scheduled examinations between two sessions in an attempt to improve on the solution found in step 1. Tabu search (Rayward-Smith et al. 1996) is a strategy for guiding local search approaches to overcome local optimality. A tabu list is maintained, which is a list of moves that have been made in the recent past of the search, and which are “tabu” or forbidden for a certain number of iterations. As such, this prevents cycling from occurring while at the same time promoting a more diversified search of the solution.


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Given a solution, tabu operators are used to search for neighborhood solutions. In the context of ETTP, it may not be easy to find an operator that could easily ensure the feasibility of the solution after the operation is performed. In this study, our tabu operator comprises an exchange of two arbitrary examination sessions on a given solution. Figure 2 illustrates an example of the swapping of two examination sessions.

Before:

Session 1 Exam A Exam B Exam C

Session 2 Exam D Exam E

Session 3 Exam F Exam G Exam H

Session 4 Exam I

Session 5 Exam J Exam K

Exchanging session 2 and session 4 After:

Session 1 Exam A Exam B Exam C

Session 2 Exam I

Session 3 Exam F Exam G Exam H

Session 4 Exam D Exam E

Session 5 Exam J Exam K

Figure 2. Session Exchange Operation This process terminates when it reaches a predefined number of non-improving iterations. The tabu search algorithm is described as follows (Algorithm 2): 1.

Given a. A feasible solution S b. Number of iterations to quit, E, i.e., after E non-improving iterations c. Tabu list size d. A set of tabu operators O

2.

For each operator in O, obtain its best neighborhood move.

3.

From the set of best moves, get the best non-tabu move, b from O.

4.

Use b to update S.

5.

Set b to be tabu.

6.

Repeat step 2 to 5 until E is reached.

7.

Return the best solution found. Algorithm 2. Tabu Search

Step 3: Selection Algorithm with Consistency Check At this final stage, we wish to convert the solution derived from the first two steps into one that is feasible. To do this, we employ a selection algorithm with consistency check. Basically, the selection algorithm selects an examination to schedule based on some criteria. The algorithm then attempts to place the exam into the timeslot determined by the initial solution, then performs a consistency check. Should there be a constraint conflict, the examination is instead placed in a feasible slot on a first-fit basis. We employ a selection algorithm named the variable ordering method (VOM), which makes use of the AC-3 algorithm for its arc consistency check. The basic VOM algorithm is as follows (Algorithm 3):

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1.

Let Q be a priority queue of all examinations sorted by some ordering strategy.

2.

Let U be a set that contains examinations that cannot be assigned a timeslot and initialize it to be empty.

3.

While Q is not empty a. Dequeue exam e in Q b. Find the first available timeslot for exam e c. If the timeslot is found then i. Assign exam e to the found timeslot Else ii. Add exam e to the U Return U

4.

Algorithm 3. Variable Ordering Method The ordering strategy of this method is based on a weighted sum of three measures. Each examination has a weight computed as follows: ExamWeight = "Measure1 + $Measure2 + (Measure3 where

"+$+(=1

Measure1 is based on the number of candidates taking this examination. Measure2 is based on the constraint degree of this examination (i.e., the number of other papers affected by the scheduling of this paper) Measure3 is based on the number of timeslots that cannot be used for scheduling this examination, due to one or more constraint conflicts.

In Measure1, we give a higher weight to large exams due to their high likelihood of violating venue capacity constraints. Measure2 attempts to schedule highly constrained exams first since they are more likely to result in constraint conflicts. Measure3 reasons that with fewer available slots for the exam, it would be prudent to schedule it first. While it may be possible to include more or fewer measures, experimentation with these criteria gave satisfactory results.

RESULTS AND ANALYSES Test Data In order to test our proposed heuristics, we obtained the student registration data from the NUS Computer Center for both semesters of the academic years 1997/1998 and 1998/1999. The data consisted of a set of text files containing the list of studentexamination tuples. Table 1 shows some of the statistics on the data given. Table 2 gives the list of examination venues that can be used for scheduling the examinations. The IMM Exhibition Hall is a commercial building not owned by NUS, and must be rented for the examinations. It should be noted that the data we acquired from the Computer Center contained some anomalies of students taking an illogical number of examinations. In particular, there were instances of students taking more than 20 examinations in one semester. We suspect that this is due to the inclusion of non-examinable and/or exempted subjects in the database. As there was no convenient way to remove these cases, they remained in our test data. Furthermore, some examinations in the manual timetable were scheduled in small classrooms and laboratories and these alternatives sites are not used in our simulation. Some extra constraints have been added based on the assumption that students normally take courses with a course code similar to their main curriculum. For example, first-year students from the School of Computing will normally take up courses CS1101, CS1102, CS1103, CS1104, and CS1105 in the their first academic year. With this in mind, we included constraints to make sure that examinations with similar course codes are at least x sessions apart. The reason for this is to allow the students to have sufficient time for rest and preparation between exams. Hence, we grouped courses with similar codes together and added these extra constraints for every group. Since there are about 2,000 extra constraints to be added for each test case, to prevent our timetabling problem from being unreasonably difficult, we decided to randomly specify 80% of the extra constraints to be soft and the rest to be hard. Table 3 shows the number and percentage of hard/soft extra constraints added to our test sets. 2001 — Twenty-Second International Conference on Information Systems

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Table 1. Statistics of Examination Test Data 1997/1998 Academic Year

1998/1999

No. of Students

Sem I 20596

Sem II 20036

Sem I 21607

Sem II 21591

No. of Examinations

1586

1603

1561

1545

No. of Student-Exam Tuples

96987

88330

101197

93693

No. of Time Conflict Constraints

22745

22582

23751

24424

245

238

277

285

1.81%

1.76%

1.95%

2.05%

Max Degree of Time Conflicts Connectivity of Constraint Graph

Table 2. A List of Examination Venue Venue IMM Exhibition Hall

Alias IMM

Capacity 1600

Gymnasium

GYM

312

Multipurpose Sports Hall 1

MPH1

750

Multipurpose Sports Hall 2

MPH2

850

CH

396

Competition Hall

Table 3. Statistics on the Dummy Constraints Added to Test Data 1997/1998 Academic Year Hard Extra Constraints added Soft Extra Constraints added

Sem I 452 (20.8%) 1718 (79.2%)

Sem II 463 (21.0%) 1739 (79.0%)

1998/1999 Sem I 346 (20.9%) 1312 (79.1%)

Sem II 365 (21.0%) 1372 (79.0%)

Experimental Details In these experiments, we compare the variable ordering method (VOM) with the combined method (CM). In particular, two search heuristics, namely the graph coloring method (GCM) and the genetic algorithm (GA), are used together with the combined method independently, denoted by CM(GCM) and CM(GA) respectively. Both methods include a post-optimization step employing tabu search, where the size of tabu list and the maximum number of non-improving moves are set to 20 and 100 respectively. All the heuristics are implemented in Java Development Kit 1.2 using IBM VisualAge TM for Java. These experiments were carried out on a Pentium III 500 PC. All computational times are rounded to the nearest second. In VOM, the values for α, β, and γ for the three measures are empirically set to 0.4, 0.3, and 0.3 respectively. In GCM, the value of LIMIT that determines the maximum number of non-improving iterations is set to 20. In GA, we use the traditional GA operators, which include the crossover and the mutation operators. Here, mutation is performed at a rate of 1%. The size of the initial GA population and the maximum number of generations are 50 and 20 respectively. The aim of these experiments is to compare the effectiveness of each heuristic on two metrics. The first is the total number of sessions required to schedule all of the examinations. The second is the paper spread per student, which is a measure of the amount of study time each student has to prepare for the examinations (Bullnheimer 1997). The results for each semester are presented in Tables 4 through 7.

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Table 4. Results of Various Heuristics on Test Data 1997/98 Semester 1 1997/98 Semester 1 Data No. of sessions used No. of papers use pre-assigned slots No. of soft constraint violations Average day spread Average slot spread No. of 2 consecutive sessions No. of 3 consecutive sessions No. of 4 consecutive sessions No. of 5 consecutive sessions No. of > 5 consecutive sessions Total Time Taken (s)

VOM No TS TS 33 33 1586 1571 463 269 1.95 2.85 3.16 4.53 10390 9431 3511 1239 1110 402 435 67 1254 51 234 504

CM(GA) No TS TS 32 32 471 713 361 300 2.77 2.96 4.47 4.78 11317 9865 2196 2212 445 314 134 66 73 55 639 527

CM(GCM) No TS TS 32 32 827 1320 373 206 2.48 3.06 4.05 4.90 10263 9426 2316 1283 737 451 1078 30 245 28 247 298

Table 5. Results of Various Heuristics on Test Data 1997/98 Semester 2

1997/1998 Semester 2 Data No. of sessions used No. of papers use pre-assigned slots No. of soft constraint violations Average day spread Average slot spread No. of 2 consecutive sessions No. of 3 consecutive sessions No. of 4 consecutive sessions No. of 5 consecutive sessions No. of > 5 consecutive sessions Total Time Taken (s)

VOM No TS TS 33 33 1603 1560 423 303 2.06 2.79 3.34 4.45 9848 9238 3421 2269 1177 270 335 216 351 34 193 416

CM(GA) No TS TS 30 32 817 759 339 345 2.52 2.55 4.05 4.13 9325 9449 1851 1508 693 260 179 129 18 43 448 481

CM(GCM) No TS TS 30 30 1010 1369 365 225 2.59 3.01 4.20 4.88 9999 8941 2729 1035 786 31 169 10 20 1 202 234



VOM No TS TS 35 35 1561 1495 370 250 2.08 2.94 3.39 4.69 10833 11089 4295 1628 859 446 356 299 1199 121 161 352

CM(GA) No TS TS 33 33 593 610 284 263 2.76 2.82 4.48 4.57 11319 9593 2677 1309 436 846 126 290 68 50 428 451

CM(GCM) No TS TS 33 33 1154 1395 245 150 2.57 3.35 4.14 5.37 11569 9418 1970 844 470 89 98 67 1019 56 176 199


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VOM No TS TS 37 37 1545 1463 340 219 2.23 2.88 3.61 4.60 11389 9077 2203 1197 1127 291 403 464 366 72 176 381

CM(GA) No TS TS 32 32 1034 1343 230 157 2.80 3.16 4.53 5.10 8528 7661 1550 990 264 173 22 15 41 2 430 461

CM(GCM) No TS TS 32 33 1287 1302 293 269 2.57 2.59 4.12 4.14 10547 9565 2609 1541 765 479 43 221 87 49 187 210

Discussion In a problem like the ETTP, one of the most important metrics to minimize is the number of sessions needed to schedule all the examinations. Especially with regard to commercial sites like the IMM Exhibition Hall, a reduction in the examination period can translate to a direct monetary savings. Table 8 summarizes and compares the number of sessions required by each of the heuristics for each of the test data used. Table 8. Number of Sessions Needed for Various Heuristics 1997/98 Sem1 1997/98 Sem2 1998/99 Sem1 1998/99 Sem2

VOM 33 33 35 37

CM(GCM) 32 30 33 32

CM(GA) 32 30 33 32

As we can see, both the combined methods consistently perform much better than the variable ordering method. One reason for this is that the VOM makes rather local decisions when deciding the ordering of the papers. That is, it makes a decision based on the status at that point in time. In contrast, CMs first make a global decision on where each examination should be assigned and then use the VOM to assist in assigning the examinations. In this sense, CM may formulate a better strategy and thus performs well. However, by doing this additional step, CM could take a longer time to generate a solution depending mainly on the search heuristic used. Next, we would like to find out how adaptable combined method is when different search heuristics are used. Here, we make comparisons based on the two search heuristics used. From Tables 4 through 7, we see that the number of examinations using pre-assigned timeslots for CM(GA) is generally much fewer than CM(GCM). One explanation for this may be that our GA uses traditional operators such as crossover, which may create more conflicts after generations. Another possible reason is that our heuristics only deal with the two critical constraints and some dummy session-apart constraints, which can be handled quite well by the GCM. However, it is surprising to see that CM(GA) maintains the quality of the initial GA solution and also performs comparably to CM(GCM) in spite of the low ratio on the usage of pre-assigned timeslots. One reasonable explanation for this is that VOM used in both the CMs did a relatively good job when making decisions in its selection of examinations to be next scheduled as well as assigning timeslots when the pre-assigned one is not feasible. This may imply that CM adapts well even when the solution generated by the search heuristic is of moderate quality. We now examine our heuristics for maximizing the study time of the students. From the experimental results, we find that CM outperforms VOM in all cases. The inclusion of the post-optimizing tabu search further improves the solutions returned. It is also observed that in our attempts to reduce the number of back-to-back conflicts or the number of students taking two consecutive sessions, the number of students taking three or more consecutive sessions and the number of soft constraint violations also seems 10



to fall. One explanation for this is that when we remove back-to-back conflicts, higher order consecutive conflicts are likely to be disintegrated into lower order consecutive conflicts. In addition, when the number of consecutive conflicts is reduced, it follows logically that the average paper spread of all students will increase. Moreover, since the extra soft constraints introduced in our test sets are session-apart constraints, a timetable with good spread generally implies that the timetable is likely to have fewer session-apart constraint violations.

CONCLUSIONS The combination of search heuristics and constraint satisfaction techniques promises to produce results that are better than any one technique used alone for the constraint satisfaction optimization problem. In this study, we proposed a hybrid-based framework for the CSOP, which is demonstrated using real-life data of a university in an examination-scheduling problem. Our combined method framework proved to be robust and flexible, producing results that were able to retain the desirable properties of solutions produced by optimization heuristics while at the same time maintaining the feasibility of the solution. We feel that the combined method can be used for almost any constraint satisfaction optimization problem, including scheduling and timetabling tasks. Its flexibility allows the user to select the algorithms that are most appropriate for the task at hand to produce the best results.

References Bullnheimer, B. “An Examination Scheduling Model to Maximize Students’ Study Time,” Practice and Theory of Automated Timetabling II: Second International Conference, Patat'97 , E. Burke and M. Carter (eds.), Toronto, 1997, pp. 78-91. Fu, Z., Lim, A., and Zhang, J. “The University Exam Scheduling Problem,” Artificial Intelligence and Soft Computing (ASC'2000), Banff, Alberta, Canada, July 24-26, 2000, pp. 350-354. Garey, M. R., and Johnson, D. S. Computers and Intractability: A Guide to the Theory of NP-Completeness, W. H. Freeman, New York, 1979. Hertz, A., and Werra, D. D. “The Tabu Search Metaheuristics: How We Used It,” Annals of Mathematics and Artificial Intelligence (1), 1991, pp. pp. 111-121. Mackworth, A. K. “Consistency in Networks of Relations,” Artificial Intelligence (8), 1977, pp. 88-119. Marin, H. T. Combinations of GA and CSP Strategies for Solving the Examination Timetabling Problem, Ph.D. Thesis, Instiuto Technologico y de Estudios Superiores de Monterrey, 1998. Rayward-Smith, V. J., Osman, I. H., Reeves, C. R., and Smith, G. D. “Modern Heuristic Search Methods,” Wiley, New York, 1996. Tsang, E. “Foundations of Constraint Satisfaction,” Academic Press, London, 1993.


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