AgentTime: A Distributed Multi-agent Software System for University s Timetabling

Transcription

1 AgentTime: A Distibuted Multi-agent Softwae System fo Univesity s Timetabling Eduad Babkin 1,2, Habib Adbulab 1, Tatiana Babkina 2 1 LITIS laboatoy, INSA de Rouen. Rouen, Fance. 2 State Univesity Highe School of Economics. Nizhniy Novgood, Russia. [email protected], [email protected], [email protected] Abstact. In the couse of eseaching distibuted timetabling poblems this aticle applies the multi-agent paadigm of computations and pesents a coespondent mathematical model fo univesity s timetabling poblem solution. The model takes into account dynamic natue of this poblem and individual pefeences of diffeent emote uses fo time and location of classes. In the famewok of that model authos popose an oiginal poblem-oiented algoithm of multi-agent communication. Developed algoithm is used as a foundation fo the distibuted softwae system AgentTime. Based on multi-agent JADE platfom AgentTime povides fiendly gaphical inteface fo online design of time tables fo univesities. Keywods: timetabling, multi-agent algoithms, distibuted systems. 1. Intoduction In moden society time scheduling plays the outstanding ole. Any schedule is the obligation, which enables to cay out authentic planning of activities fo a sepaate peson, and whole industial systems as well. Timetabling epesents an impotant eseach activity in the scheduling theoy, and focuses on such poblems as optimal lectue s schedules in educational institutions, week cyclic schedules of plane s flights within the famewok of seveal aipots, week o daily schedules of ailway tanspotation, etc. Fo all these poblems the inteval of time, inside which the given set of jobs should be fulfilled, is known befoehand. Thus, the minimum of the schedule s length is not usually consideed as a pimay goal, - othe citeia ae used fo estimation of quality of the schedule having been built. Fo example, in an educational institution the timetable design pocess should achieve the following goals: minimization of maximal length of a woking day, minimization of the numbe of the "holes" in the schedules of goups and pofessos, maximal satisfaction of pesonal pofesso s pefeences to the time and location of classes, etc. In the cuent situation, when many educational institutions apidly gow in size, and distibution scale, wide application of effective softwae systems fo distibuted solution of timetabling poblems becomes vey impotant.

2 At pesent thee ae many vaious algoithms fo dawing up the time tables in univesities. The fundamental appoaches ae based on the well-known linea and intege pogamming paadigms [2, 3, 4, 5, 6]. Howeve, seveal eseaches of 80s have shown that the intege pogamming is not equally effective fom the point of view of the calculations volume. The high computational costs make intege pogamming pooly attactive to the lage tasks of timetable design, because that method does not guaantee poductivity, when the sizes and complexity of tasks gow [3, 7]. Last twenty yeas have shown the inceased inteest of the eseaches to development of the appoaches fo the design of timetables with use of vaious metaheuistics [3], like simulated annealing, Tabu Seach, genetic algoithms (GAs), and thei hybids [8, 9, 10, 11, 12, 13, 14, 19]. It is affimed, that among othes, GAs have lage capacity, and allow to find the geatest numbe of the feasible solutions [15, 16, 17]. Nevetheless, when GAs ae exploited, thee ae difficulties in the desciption of contolling paametes, in definition of exact oles of cossove and mutations, as well as in analysis of convegence [18]. Also, it should be noted, that the majoity of the consideed appoaches follow the paadigm of centalized systems, they do not allow the emote uses goven the pocess of timetable design. In complex distibuted and evolving systems like moden vitual univesities and pee-to-pee communities, that shotcoming makes impactical classical methods, and demands new timetabling pinciples, which take account of eal-time use s pefeences in complex changing envionments. A multi-agent appoach epesents a successful paadigm fo those kinds of poblems, when an optimal o quasi-optimal solution is built in the esult of inteaction of lage numbe of autonomous computational entities. In geneal, fo timetabling applications seveal types of multi-agent algoithms ae suitable. The fist type of algoithms includes economics-based models of inteaction [1, 20, 21]. The second one consists of vaious geneic algoithms fo solution of Distibuted Constaint Satisfaction Poblems (DCSP) [22, 23]. But the most effective algoithms, compising the thid type, wee specifically designed fo a paticula scheduling poblem. Such specialized algoithms apply all domain- o poblem-specific infomation and show unbeaten poductivity. The esults ae known [24, 25, 26, 27], whee authos popose poblem-specific algoithms of agent s inteaction fo the meeting scheduling. Although such algoithms fit well the timetabling model, and have attactive computational efficiency, thei diect application fo univesity s timetabling is not so staightfowad and equies additional effots. In the given aticle authos popose new multi-agent algoithm fo univesity s timetabling, and descibe basic pinciples of the distibuted system AgentTime, based on that algoithm. Pesentation of the esults has the following stuctue. Section 2 descibes the mathematical model of univesity s timetabling, which includes use s pefeences. Section 3 gives oveview of the coesponding multi-agent algoithm fo the design of timetables. In section 4 cetain topics of AgentTime s softwae implementation ae consideed. Oveview of esults and discussion ae pesented in Section 5. Section 6 contains efeences.

3 2. The Poposed Mathematical Model Fo the Univesity s Timetabling Poblem The exact mathematical statement of the univesity s timetabling poblem foms the basis of ou own multi-agent algoithm fo design of the educational schedule. Fo the sake of geneality we use the tem teache to denote diffeent kinds of univesity employees (e.g. pofessos, instuctos, etc), the tem steam to denote a steam, and the tem subject to denote diffeent kinds of student s subjects. Also we give the same name of use to all of the stakeholdes of the schedule (e.g. the teaches and student s goups). In ou mathematical model we will use the following designations. Student s goups and Steams. g G the unique identifie of the goup. G the set of goup s identifies. G = γ the total numbe of goups. Each goup belongs to one steam at least. Some steams can consist of a single goup, but in most cases seveal goups fom a steam with the following constaints: 1. All goups of the same steam exploit the same classooms fo lectues. 2. Lectues ae deliveed to all goups of the steam at the same time. 3. Each steam has as minimum one lesson. R the set of steam s identifies. R = ρ the total numbe of steams. R the unique identifie of the steam. Each single goup can be teated as a sepaate steam, thus ρ γ. C G the steam. C= { C1, C2,..., C ρ} the set of steams. Teaches. P the set of unique teache s identifies. p P the unique teache s identifie. Timetable uses. Union of the goup s set and the teache s set gives us the set of the timetable uses: M = G P, m M the unique identifie of the timetable use. Time. W the set of the days of the week. w W the cetain day of the week. g = { 1, 2,...,7} J { 1, 2,...,8} the lesson s numbe. ( ) W W the set of leaning days fo the goup g, j = g { w, j w, j } T= W J the set of timeslots, which ae the elementay units in the timetabling poblem. Fo example, the timeslot (1, 2) means the second lesson on Monday. Fo each timetable use m the set of fee timeslots T + m T is known. The set of denied timeslots T m T is known also. We assume the obvious constaints ae tue (i.e. + + Tm Tm = T; Tm T m = ). Subjects. In ou model teaches conduct lectues and manage pactical execises. Lectues ae deliveed to the whole steam, while pactical execises ae oganized fo a single goup only. Also some pactical execises impose estictions on allowable classooms, like compute o chemisty labs. To descibe all these peculaities, let s intoduce the following mathematical stuctues.

4 = { 1, 2,..., σ } S the unique lectue s identifie; = { θ } S the set of lectue s identifies deliveed to the steam. s Q 1, 2,..., the set of pactical execise s identifies oganized to the steam. q Q the unique execise s identifie; Each lectue s assignment can be uniquely identified by a pai( s, ) RS, whee {( s, ), s } RS = R S (1) The total numbe of lectue s assignments is computed as follows: RS ρ = σ = 1 (2) Each cetain execise s assignment can be uniquely identified by a ti- g,, q RQG, whee ple ( ) {( g,, q), q, g } RQG = R Q C (3) The total numbe of execise s assignments is computed as follows: ρ RQG = C θ, whee = 1 C is the total numbe of goups in the steam C. Fo futhe analysis diffeences between lectues and pactical execises can be neglected and the united set of subjects E will be used: E= RS RQG (4) Cuiculum consists of subjects assignments fo each of the teache duing one semeste (fall) in the following fom: δ : E P δ1 ( e), e RS δ ( e) = δ2 ( e), e RQG δ : RS 1 P, whee P the set of teaches; RS the set of lectue s (5)

5 assignments. δ 2 : RQG P, RQG the set of execise s assignments. δ 1, = means, that teache 4 delives lectue 2 fo steam 1, δ = means, that teache 7 manages pactical execise 2 fo goup 4, Fo example, ( ) and ( 1, 2, 4) 7 2 included into the steam 1. Given the cuiculum δ, we can easily compute the total numbe of subjects E m assigned to the teache (o the goup) with identifie m: ( ) ( ) ( ) E = { e m P δ e = m} { e=, s m C s S } m { e=, q, m m C q Q } (6) Room s stock consists of laboatoies, lectue halls and classooms available fo subjects in the univesity. It is modeled by the set A of unique oom identifies. Fo each element of the set of subjects E, a subset of pemitted ooms A e is selected : A A e. The pimay goal of the timetabling poblem in ou model is fomulated as looking fo the feasible mapping fom the set of subjects E to the set of timeslots T: τ : Fo example, mapping ( 1, 2) ( 4, 4) E T (7) τ = means that subject 2 fo steam 1 will be given on Thusday duing the fouth lesson. Related with the mapping τ, the mapping α should assign a classoom fo each subject: α : E A, whee E the set of subjects; A the set of classooms. Fo example, mapping α ( 1,2 ) = 101 means that subject 2 fo steam 1 will be conducted in the oom 101. Constaints fo the univesity s timetabling poblem ae defined as follows. 1. The teache can conduct only one subject at the single timeslot. ( ) ( ) ( ) ( ) p P, e, e E : e e δ e = δ e = p τ e τ e (9) In one classoom only one subject can be given at the single timeslot. ( ) ( ) ( ) ( ) a A, e, e E : e e α e = α e = a τ e τ e (10) (8)

6 3. Each goup has no moe than one subject at the single timeslot. :( 1 ( 1,... ), 2 ( 2,... ) ) (..., ), (..., ) E τ τ g G e = e = E g C g C e e ( e1 g e2 g e1 e2) ( e1) ( e2) = = (11) Subject s pioity. It is obvious, not all subjects have identical impotance within the famewok of educational pocess. As such, it is necessay to set the pioity ode among diffeent subjects, so subjects with highe pioity will boow the best time and location. In ou model the pioity is modeled as the patial ode on the set of subjects E: e1 e2 U( e1) U( e2), whee e, e E ; U( e) = + k ( e) + k ( e) + k ( p ) 1 2 M e e the utility of the subject e ; Me = { m ( e= (, s) RS m C) ( e= (, q, m) RQG )} the total numbe of goups fo those the subject e is given; k1 ( e) {0,5,10} the measue of subject s impotance fo the steam (0 optional, 5 impotant in geneal, 10 impotant fo steam); k2 ( e) {0, 2} 0 undegaduate, 2 gaduate; pe = p, δ ( e) = p the teache s identifie; k3 ( pe ) {0..5} the estimation of the novelty level of the mateial given by the teache p e. (12) Use s Pefeences compise the impotant pat of ou model. Each pefeence is modeled by a numeic value fom the ange [0,1]. Value 0 coesponds to the least desied altenative, and value 1 coesponds to the most desied altenative. The model includes two kinds of the use s pefeences: pefeences of the use m M fo the time of subjects: f m m m 1 + : E T [0,1] (13) pefeences of the use m M fo the location of subjects: f 2 : EA [0,1] m m, whee m e m EA = {( ea, ) a A e E } the set of feasible pais subjectclassoom. (14)

7 We use an evident epesentation of the use s pefeences in the fom of gaphics tables (table 1, 2). The dake colo denotes the moe pefeable altenative (in espect of time o location). Table 1. Pefeences of the use m fo the desiable time of subjects, Lesson s numbe, j 1 f m, Mon Tue Days of the week, w Wed Thu Fi Sat Sun Table 2. Pefeences of the use m fo the desiable location of subjects, Classoom, a f 2 m Citeion of timetable quality genealizes seveal patial citeia, and evaluates the solution found, namely the pai of mappings τ ( e), α ( e). The fist patial citeion evaluates the sum of the use s pefeences fo the time of the subject e: F 1 τ = f 1 e τ e (15) e ( ) ( ) ( ) m M e m, max The second patial citeion evaluates the sum of the use s pefeences fo the location of the subject e: ( ) ( α) α( ) F = f e, e max 2 2 e m m M e 2 f m (16) The genealized citeion is constucted as follows: 1 2 ( τα) ( e e ) F, = F + F max e E (17)

8 The solution of the descibed poblem consists of the found mappings τ, α, assuming that all constaints ae satisfied, and the genealized citeion has a maximum value. 3. The Multi-Agent Algoithm Fo Timetable Design We took fo the basis of ou algoithm the well-known multi-agent algoithm MSRAC fo meetings scheduling by A. Ben Hassine et all. [27]. Although some coespondences still emain, ou algoithm is specifically designed fo a quite diffeent poblem of univesity s timetable design, and togethe with time schedule it gives also an occupancy schedule fo classooms. In ou algoithm we ecognize two oles of agents: agents-oganizes and agentspaticipants. The agent s stuctue also mimics the application domain, so we classify all agents as teaches, goups and classooms. Agents-teaches play the ole of oganizes; agents-goups and agents-ooms play the ole of paticipants. The numbes of agents-teaches and agents-goups coespond to the eal numbes of the teaches and the goups in the univesity. One agent-oom coesponds to all classooms in the context of the single time table. Collective seach fo the best time and location of the study involves communication between diffeent agents. Fo each study the agentteache pefoms a set of actions, compising the following state diagam (fig. 1). Fig. 1. The state diagam of the agent-teache s algoithm.

9 The agent-teache pefoms state tansitions in accodance with the desciption given below. 1) Ask_when_avail. That is the fist state in the algoithm. The agent-teache sends to all agents of goups the quey WHENAVAIL with the study s identifie, equesting available time fo that study. The agents of goups answe by the message USERAVAIL, in which they infom when the agent is fee, and has available time fo the study. If all agents have infomed the answe, then the agent-teache finds intesection on time. If the intesection is empty, then the agent comes the final state imposs_meeting. 1a) Imposs_meeting. In that state the agent-teache founds itself if intesection of available fo othe agents times is empty and the total solution was failed. The study is maked as having no solution. 2) Ask_subj_pefs. In that state the agent-teache equests pefeences fo time and location (the message EVALUATE). The agents of goups eply own pefeences in the message SUBJPREFS. The agent-teache sots eceived pefeences fo time and fo location in accodance with citeia (15) and (16). 3) Popose_time. The agent-teache selects the fist timeslot fom the odeed list of the pefeences, and sends it along with the study s identifie to the agents of goups inside the message TIMEPROPOSAL. In esponse the agents of goups analyze own agendas. If the poposed timeslot is fee in the agent s agenda, the agent gives the positive answe, sending the message ACCEPT. Else the agent compaes the pioity of the study in the agenda with the pioity of the study in the message. If the pioity of the message s study is geate, then the agent accepts new poposal and sends the message ACCEPT. In the opposite case the agent sends the message REJECT. The agent can apply the metopolis citeion [27] fo decision making when the pioities ae equal. In the case of total acceptance of the poposed timeslot, the agent-teache passes to the next state Popose_location; in default the agent emains in the state popose_time, and chooses the next timeslot to negotiate. If all timeslots wee ejected, it means that the decision fo the cuently selected study does not exist, and the agent-teache passes to the state (3а) Solnot_found. 3a) Solnot_found. In that state the agent-teache founds itself if all poposed fo timeslots wee ejected by the agents of goups, and the total solution was failed. The study is maked as having no solution. 4) Popose_location. The agent-teache sends the soted list of classooms to the agent of classooms inside the message LOCPROPOSAL. That message contains also the study s identifie and the timeslot s identifie. Using own occupancy list, the agent of classooms seaches fo the fist classoom in the list, which is available fo the timeslot given o occupied by the study with a lowe pioity. If the seach was successful, and the classoom is found, in eply to the agent-teache the agent of classooms sends the message ACCEPT with the identifie of the oom found. In the failue case the agent of classooms sends the eply REJECT. Once the positive ACCEPT eply is eceived, the agent-teache moves to the next state. In the esult of REJECT eceiving the agent etuns to the state popose_time fo selecting the available timeslot fo the study. 5) Fix_meeting. If the agent-teache occus in that state, it means that both the timeslot and the classoom fo the study wee successfully found. In the esult the agent-teache sends to all othe agents the message FIXMEETING with the identifies

10 of the study, the timeslot, and the classoom. If the agent-goup does not have assignment fo the eceived timeslot, the timeslot is fixed. By a simila way the agent of classooms fixes the location. If the eceived timeslot (o the classoom) is occupied, the agent discads assignment of the study with lowe pioity, and sends to the agentteache the message CANCEL MEETING, which is fowaded futhe to othe agents in ode to modify thei agendas. In the states (1), (2), (3) and (4), if some agents did not send the answe duing a pedefined time peiod, the agent-teache places the study, being unde consideation, into the list of the cancelled subjects, to ety attempts late. Once all agents-teaches finish state pocessing, the common schedule is consideed to be complete. One impotant featue of ou algoithm is that of the patial timetable is always available. The complete timetable, including all the subjects, sometimes simply does not exist. In such a case, howeve, the consideed algoithm will build the consistent time table, with some subjects of low pioity ignoed. 4. Implementation Details of the Softwae System AgentTime The descibed mathematical model and the multi-agent algoithm wee applied in the couse of design and development the softwae system fo time tabling called Agent- Time. AgentTime uses ich communication and agent-life cycle capabilities of Javabased JADE multi-agent platfom [29], and has highly distibuted softwae achitectue (fig.2). Flexible multi-tie achitectue of the system suppots simultaneously multiple timetable design sessions and inteaction of multiple agents. Fig. 2. The softwae achitectue of AgentTime timetabling system.

11 In AgentTime agents indiectly communicate with each othe by passing the messages in accodance with a poblem-specific ontology (table 3). Table 3. Multi-agent ontology fo timetable design Message Semantics WHENAVAIL Inquiy to the agent-goup fo available timeslots. USERAVAIL Agent s esponse to the message WHENAVAIL. The message contains the vecto with available timeslots. Fomat : (a 11 a 12 a 18 a 21 a 22 a 28 a 71 a 72 a 78 ), whee a ij {0,1}, 1 woking day, 0 weekend. IMPOSSMEETING The message to the seve agent about impossibility to find a time table fo the subject with id sbj_id. Fomat: (sbj_id). EVALUATE Inquiy to the agent of goup fo time and location pefeences elated with the subject sbj_id. Fomat:(sbj_id). SUBJPREFS Agent s esponse to the EVALUATE. Fomat: (sbj_id (w 11 w 12 w 18 w 21 w 22 w 28 w 71 w 72 w 78 ) ((L 1 p 1 ) (L 2 p 2 ) (L n p n ))), whee 0 wij 1( i = 1..7, j = 1..8) evaluation of i -th day of week and, j -th lesson; 0 Lk 1, k = 1, n the numbe of the classoom; missing classooms have the pioity with value 0; 0 pk 1,0 k n the pefeence of the classoom L k. TIMEPROPOSAL The agent-teache poposes time fo the subject sbj_id. Fomat: (sbj_id (d p)), whee d the day of the week; p the numbe of the lesson. LOCPROPOSAL The agent-teache poposes location fo the subject sbj_id. Fomat: (sbj_id (L 1 L 2 L m )), whee Lk, k = 1, m the identifie of the classoom. The classooms ae soted in accodance with the pefeences. ACCEPT/REJECT Agent s esponse to the message TIMEPROPOSAL (ACCEPT o REJECT). If the poposal is accepted the message contains the classoom s identifie. Fomat: (L), whee L is the id of classoom. FIXMEETING Inquiy to fix the timeslot and location fo a cetain subject sbj_id. Fomat: (sbj_id (d p) L), whee d the day of week; p the id of the lesson; L the id of the classoom. CANCELMEETING Notification about cancelling a conflicting subject. Fomat: (sbj_id (d p) L), whee d the day of the week; p the id of the lesson; L the id of the classoom.

12 Inteaction of the agents duing the design of timetable can be illustated by the UML sequence diagam in fig. 3. In AgentTime apat fom peviously mentioned types of the agents we use the dedicated SeveAgent which is esponsible fo communication with extenal data souces, logging and othe technical tasks. Fig. 3. The state diagam of the agent-teache s algoithm. Diffeent uses of AgentTime can inteact with the system using diffeent end-use tools, including web-bowses and PDAs. The mostly used way of inteaction assumes application of applet-based gaphical intefaces (fig.4), but also JSP-based intefaces ae available.

13 a) b) Fig. 4. Examples of AgentTime s gaphics intefaces: a- assignment of pefeences; b epesentation of the eady timetable.

14 5. Discussion This aticle focused on the impotant poblem of time tables design fo educational institutions. To tackle this poblem in the context of moden distibuted and highly dynamic univesities we poposed the mathematical model and coespondent multiagent algoithm fo iteative timetabling in pesence of diffeent subjective pefeences fo time and location of subjects. The theoetical consideations become a foundation fo development of the multi-agent softwae system AgentTime. That system facilitates distibuted time planning and allocation of timeslots and classooms. The developed algoithm belongs to the class of domain-specific multi-agent algoithms and shows good pefomance metics. Analysis shows that in the case of single computational node computational complexity of the algoithm C fo allocation of timeslots and ooms can be estimated as follows: C S log S T n, n, whee 2 0( g l ) S is the numbe of subjects, 0 ( g, l ) T n n - a constant detemined by the poblem s conditions. If AgentTime is distibuted among P S computational nodes, C S then estimation of pocessing time t p will be t P log2s T0 P = P. In the exteme case, when P= S, t p will not be geate then T0 log2 S. Compaing ou esults with othe known appoaches to multi-agent timetabling like the algoithm MSRAC [27], we can note that ou system is capable of solving a moe geneal poblem, allocating not only timeslots, but classooms also. With a few modifications poposed model and algoithm will be suitable fo managing othe impotant esouces as well. At the same time we need to impove theoetical backgound of ou algoithm to igoously pove the optimality of the solutions found in tems of the citeia (16) and (17). In the neaest time we ae going to pefom wide-aea field expeiments with AgentTime to test its obustness and quality of timetabling in eal conditions of the complex univesity. We ae also inteested in extending the poposed mathematical model and softwae implementation of AgentTime with othe appoaches to multiagent coodination. In this context application of the paadigm Contolle-Vaiable Agent [28] is seemed to be vey pomising. This wok was patially suppoted by Russian Fund of Basic Reseaches (gant # ). 6. Refeences 1. Cheng J.Q., Wellman M.P. The WALRAS Algoithm: A Convegent Distibuted Implementation of Geneal Equilibium Outcomes // Jounal of Computational Economics, pp Sandhu K.S. Automating Class Schedule Geneation in the Context of a Univesity Timetabling Infomation System. PhD Thesis

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