Ant Systems (AS) is a recently proposed meta-heuristic inspired on biological behaviors, which has been applied to a variety of combinatorial optimization problems, including the QAP (Quadratic Assignment Problems). In this work, we have studied the adaptation of the Ant Systems meta-heuristic to two different real-life assignment problems, which appear in educational institutions: the timetabling problem (assigning courses to classrooms and times), and the assignment of final proyects to students. There is no standard definition for these problems, as in each institution the rules and objectives are different. We have been successful in adapting. AS to tackle these problems as defined by our institution rules, showing the adaptability of this meta-heuristic to complex, real-life problems. In both cases the AS meta-heuristic obtained good quality solutions (in the timetabling case, at the cost of longer running times)

Cancela, Héctor

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Applying Ant Systems to two real-life assignment problemsHector Cancelacancela@ng.edu.uyInvestigacion Operativa - InCo - Pedeciba InformaticaFacultad de Ingeniera, Universidad de la RepublicaMontevideo, UruguayReporte Tecnico INCO 00.04Mayo 2000AbstractAnt Systems (AS) is a recently proposed meta-heuristic inspired on biological behaviors, whichhas been applied to a variety of combinatorial optimization problems, including the QAP (QuadraticAssignment Problem).In this work, we have studied the adaptation of the Ant Systems meta-heuristic to two dierentreal-life assignment problems, which appear in educational institutions: the timetabling problem(assigning courses to classrooms and times), and the assignment of nal projects to students. Thereis no standard denition for these problems, as in each institution the rules and objectives aredierent. We have been successful in adapting AS to tackle these problems as dened by ourinstitution rules, showing the adaptability of this meta-heuristic to complex, real-life problems. Inboth cases the AS meta-heuristic obtained good quality solutions (in the timetabling case, at thecost of longer running times).Key words | Combinatorial Optimization, Meta-heuristics, Assignment.1 IntroductionThe objective of this work is to study the application of the Ant Systems meta-heuristic to tworeal-life assignment problems. This meta-heuristic has been developed recently, and most of theliterature concerns its application to benchmark combinatorial optimization problems such as theTSP, the VRP, the SOP, the QAP; but very little research seems to have been done on its applicationto real-life combinatorial problems, leaving it open the question of its applicability in more generalcontexts.The paper is organized as follows. Section 2 presents the Ant Systems meta-heuristic. Section 3presents the two problems to be studied, a timetabling problem and a project assignment problemoccurring at the Facultad de Ingenieria of the UDELAR (Universidad de la Republica, Uruguay).Section 4 discusses some aspects of the adaptation of Ant Systems to the previously mentioned real-life problems, and presents some results of the experiments based on the implementation proposed.Finally, the conclusions of this work are presented in Section 5.2 The Ant Systems meta-heuristicThe Ant Systems meta-heuristic is a recently developed approach to combinatorial optimizationproblems, which is based on the observation of the behavior of a very common and successful1biological system, namely ant colonies. In particular, when individual ants search for food in theneighborhood of their nest, they deposit a chemical substance, called a pheromone. Other antscan detect this substance with their antennae. In the absence of pheromone, the ants exploresthe surrounding area in a random manner; but if pheromone trails exist, the ants will follow the\stronger" trails with large probability. This gives rise to a positive feedback eect, as these newants will reinforce the existing paths with their own pheromone trails.The indirect communication arising from this behavior allows to share the experience betweenall individual ants, and to nd good paths between the nest and the food sources, even if each anthas an extremely limited view of the surrounding area.These natural systems inspired Colorni, Dorigo and Maniezzo [CDM92] to design a new heuristic,based on the idea of considering articial ant processes which build solutions to the optimizationproblem, and share their experience by means of an articial pheromone trail, in eect a numericalcommon memory which can be read and modied by the ant processes to reect the use of buildingblocks in the problem solution. Since then, a number of changes and extensions have been proposedto the original model, to improve its eciency and to solve its most important shortcomings.The Ant Systems meta-heuristic has been proposed [Tai99] as a model which generalizes the rstheuristics based on ants behavior, and can take into account later extensions in a single framework.To do this, the original analogy is extended by considering the existence of a \queen" process; asthe queen of the ant nest creates new ants, determines their type, and generally co-ordinates thewhole colony, the articial queen process will co-ordinate all articial ants processes.Following [Tai99], we consider that the model is based on the following three analogies: Real ants correspond to processes in charge of building solutions to the combinatorial problemconsidered; these processes are often referred as Articial ants, or Ant processes. The pheromone trails correspond to a common memory, recording a value associated witheach component of a solution, that is updated on the basis of the solutions built by the Antprocesses. The queen corresponds to a central process in charge of activating and co-ordinating Antprocesses, and of managing the common memory.An Ant System is described by a set of Ant processes that receive problem data, memory state,and other parameters from the Queen process; build a new solution probabilistically, and sendthe new solution to the Queen process; and a Queen process wich initializes the memory, andthen iterates choosing parameters to create new Ant processes, receiving the solutions built by theprocesses and updating the memory, until a stopping criterion is met; in which case returns the bestsolutions produced by the system.To apply the Ant System meta-heuristic to any given combinatorial problem, it is necessary todene: 1) The type of information memorized. 2) The memory update rules applied by the Queenprocess. 3) The constructive procedure used by the Ant processes.3 The assignment problemsThe Ant Systems literature has been mainly concerned with the application of this meta-heuristicto a number of benchmark combinatorial problems, such as the TSP[DG97] and VRP [GTA99],the QAP [Stu99], the SOP [GD97], and the Job Shop Scheduling problem [CDMT94], with good2results. Nevertheless, two common features among these problems is that they have a simpleobjective function, and that the domain constraints are (relatively) few. In real-life problems, thesituation is generaly the inverse, as we may have more than one objective, and a big number ofdierent domain constraints, which must be taken into account.We have studied the application of Ant Systems to the solution of two real-life assignmentproblems which are present in our University (as in many other educational institutions). The rstone is the timetabling problem, consisting in assigning classrooms and times to the dierent courses;the second one is the assignment of nal projects to students.Problems of the timetabling kind have received considerable attention in the literature (see forexample [Her91, DS97, de 97]); there are many dierent formulations, which vary in the constraintsand objectives. The rst step in our work was to analyze how our University (implicitly) denedthis problem; the main points are the following: There are a number of classrooms, with dierent capacity; the classrooms can be used fromMonday to Saturday, from 08:00 to 24:00. There are a number of titulations (degrees); each degree is organized in semesters, there beinga number of courses corresponding to the degrees in each semester. A course can be sharedamong dierent titulations; also, a course can be divided in groups, when the number ofprospective students is too big. Each group should receive a given number of weekly lessons, of given duration (for example,two lessons of 90 minutes; or three lessons of 120 minutes, etc.). The students are enrolled in the titulations, and we know (approximately) the number ofstudents in each semester for each titulation.Based on these entities, there are a number of \hard" constraints, which must be satised: eachgroup of each course must be assigned a classroom for each weekly lesson, with capacity at leastequal to the expected number of students; a classroom can not be assigned to two groups at thesame time; two courses corresponding to the same semester of the same titulation can not haveany of their weekly lessons assigned at the same day and time of the day. There are also other\soft" constraints, which can be relaxed: the teachers of the courses can state their preferred timeof the day; there should not be (if possible) empty time between lessons of dierent courses ofthe same semester of the same titulation; the weekly lessons of a course should be \uniformelydistributed" in the week if possible (for example, two weekly lessons on Tuesday and Thursday arebetter distributed than on Monday and Tuesday, or on Monday and Friday). The objective is tond a feasible solution, or at least a solution which satises all hard constraints and \least violates"all soft ones.A typical problem size for the Faculty of Engineering involves 8 titulations, 10 semesters pertitulation, between 3 and 8 courses per titulation per semester, 30 classrooms and 5000 students.In our case we worked with a subset of the problem, involving only the most dicult task, namelythe courses of the rst semester of all titulations, which involves 4 courses (divided in 26 groups)and about 1200 students.The second problem is less complex, both in the number and type of entities involved, and inthe constraints to be taken into account. It consists in assigning (groups of) students to projects.The entities involved are: 3 Projects: each project is proposed by a teacher, has a minimum and maximum group size,and a minimum and a maximum number of groups involved. The teachers may state theirpreferences for the students involved. Students: each student can state his preferences for each of the proposed projects; and also canstate his anity for working with other fellow students in each of the proposed projects. Theanities need not be coherent over the projects (student A may want to work with studentB for project P1 but with student C in project P2), and need not reciprocal (student A maywant to work with student B, who in his turn wants to work with C).The constraints are that each student must be assigned to a project and that each project musteither be assigned a number of groups between the minimum and maximum stated (each groupcomposed by a number of students between the given minimum and maximum group size) or notbe assigned at all. The objective is to maximize the satisfaction of both teachers and students.An interesting point of this problem is the fact that the restrictions are stated in term of groupsof students, but the assignment in itself is at the student level (not at the group one); this preventsit from being easily classied among standard assignment problems.A typical problem size for the Computer Engineering degree involves about 40 projects and 100students, most projects involving one or two groups of between 1 and 3 students.4 Implementation and experimentationThe following step in this work consisted in adapting the AS meta-heuristic for the solution ofthe two problems stated in the previous section. As usual with optimization meta-heuristics, theimplementations were dierent for each problem, in order to reect its special characteristics.For the timetabling problem, the memory was dened as a three-dimensional matrix, indexed bythe lessons (belonging to a group, which in turn belongs to a course), the classrooms, and the timesof the day (discretized in half-hour intervals). For the project assignment problem, the memorywas dened as a two-dimensional matrix, indexed by the students and the groups of the projects.In both cases, a memory entry will be higher when that particular sub-assignment has appeared ingood-quality solutions.We experimented with dierent memory update rules. In particular, we compared the case whereall ants were allowed to contribute to the memory update, with the case where at each iteration,only the ant which found the best solution was taken into account in order to update the memory(this is referred to in the literature as an elitist strategy); in both problems the elistist strategy hadsuperior results. The memory update includes both a \forget" mechanism (at each iteration, oldentries are exponentially smoothed), and a \reinforcement" mechanism (in the case of the timetable,if a lesson has been assigned to a classroom at a given time of the day in the best solution, then apositive number is added to the corresponding entry in the matrix; this number is inversely relatedto the cost, in order to reinforce low-cost solutions).The usual constructive procedure employed by the Ant processes is based on the iterative build-ing of partial solutions; at each step a new element is added to the solution. The choice is prob-abilistic, guided partially by the common memory (which reects the frequency of use of partialassignments), and partially by some greedy evaluation function (usually called the visibility). Inour case, the elements of the solution are partial assignments (of a lesson to a classroom at a given4time, or of a student to a group of a given project). The evaluation function takes into account thesuitability of this partial assignment, independently from the rest of the solution.Two parameters control the relative weights of the memory and the greedy function. Tere isno general procedure to x their values, which are often chosen empirically. We also experimentedwith the sensibility of the AS meta-heuristic with respect to these parameters, obtaining in bothproblems results similar to the ones published for benchmark problems in the existing literature.The implementations were run both on \toy" problems, to debug the programming and adjustsome parameters of the algorithm, and on real data corresponding to a particular semester. Inthe case of the project assignment problem, the AS meta-heuristic was very successful, obtainingin some minutes solutions which used to take many hours of a skilled person work. In the caseof the timetabling problem, we also obtained feasible solutions in few iterations; but enforcingthe \soft" constraints required running times considerably higher. It is important to note thatthe solutions which are applied in the institution, and which are obtained by manually applyinginformal heuristics, usually result in the violation of some constraints and/or the redenition of theproblem (i.e., groups can be split or joined, time restrictions changed, etc.).5 ConclusionsIn this paper we have presented the AS meta-heuristic and its application to two real-life assignmentproblems, showing that this method can be applied not only to benchmark problems, but is alsouseful in complex real-life combinatorial optimization problems.Among many open questions, it would be of particular interest to further study the dependencyof the AS meta-heuristic on the parameters which control the relative inuence of the memory andthe greedy function, and if possible to dene an automatic procedure to x their values.AcknowledgmentThe author wants to thank students Marcos Carlevaro, Santiago Costabel and Fernando Rey, whichparticipated in the adaptation and implementation of the AS meta-heuristic under his direction.References[CDM92] A. Colorni, M. Dorigo, and V. Maniezzo. Distributed optimization by ant colonies.In F.J. Varela and P. Bourgine, editors, Toward a practice of autonomous systems:proceedings of the First European Conference on Articial Life (ECAL 91), pages 134{142. MIT Press, Cambridge, 1992.[CDMT94] A. Colorni, M. Dorigo, V. Maniezzo, and M. Trubian. Ant system for job-shop schedul-ing. Belgian Journal of Operations Research, Statistics and Computer Science, 34(1):39{53, 1994.[de 97] D. de Werra. The combinatorics of timetabling. EJOR, 96:504{513, 1997.[DG97] M. Dorigo and L.-M. Gambardella. Ant colony system: a cooperative learning approachto the traveling salesman problem. IEEE Trans. on Evolutionary Computation, 1:53{66,1997. 5[DS97] Andreas Drexl and Frank Salewski. Distribution requirements and compactness con-straints in school timetabling. EJOR, 102:193{214, 1997.[GD97] L.M. Gambardella and M. Dorigo. HAS-SOP: Hybrid ant system for the sequentialordering problem. Research Report IDSIA-11-97, IDSIA, 1997.[GTA99] L.M. Gambardella, E. Taillard, and G. Agazzi. MACS-VRPTW: a multiple Ant ColonySystem for Vehicle Routing problems with time windows. In D. Corne, M. Dorigo, andF. Glover, editors, New Ideas in Optimization. McGraw-Hill, 1999.[Her91] A. Hertz. Tabu search for large scale timetabling problems. EJOR, 54:39{47, 1991.[Stu99] Thomas Stutzle. ACO algorithms for the Qadratic Assignment Problem. In D. Corne,M. Dorigo, and F. Glover, editors, New Ideas in Optimization. McGraw-Hill, 1999.[Tai99] Eric D. Taillard. Ant systems. Research Report IDSIA-05-99, IDSIA, February 1999.6

Applying ant systems to two real-life assignment problems

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Applying ant systems to two real-life assignment problems

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