Optimal Solution to the Problem of Balanced Academic Curriculum Problem Using Tabu Search

Transcription

1 Optmal Soluton to the Problem of Balanced Academc Currculum Problem Usng Tabu Search Lorna V. Rosas-Téllez 1, José L. Martínez-Flores 2, and Vttoro Zanella-Palacos 1 1 Engneerng Department,Unversdad Popular Autónoma del Estado de Puebla UPAEP,21 sur 1103 Col. Santago C.P ,Puebla Pue., Méxco 2 Interdscplnary Center for Postgraduate Studes, Research, and Consultng,Unversdad Popular Autónoma del Estado de Puebla UPAEP, 17 sur 901 Col. Santago C.P ,Puebla Pue., Méxco Abstract - The Balanced Academc Currculum Problem (BACP) s a constrant satsfacton problem classfed as NP- Hard, ths problem conssts n the allocaton of courses n the perods that are part of a currculum such that the prerequstes are satsfed and the load of courses s balanced for the students. In ths paper s presented the soluton for a modfed BACP where the loads may be the same or dfferent for each one of the perods and s allowed to have some courses n a specfc perod. Ths problem s modeled as an nteger programmng problem and s proposed the use of tabu search for ts soluton because was not possble to fnd solutons for all the nstances of ths modfed problem wth formal methods. Keywords: Academc Currculum Balanced Problem, Tabu Search 1 Introducton A currculum s formed by a set of courses and these courses have assgned a number of credts that represent the effort n hours per week that the student requres to follow the courses successfully. For parents or tutors and for the nsttuton represents the economc cost of ths course. The academc load s the sum of the credts of all the courses n a gven perod. Therefore, the correct plannng of the currculum results n beneft of the all the nvolved: For the nsttutons, favors the departmentalzaton and the resultng cost savngs, for the students, one good load dstrbuton represents the academc effort that they requre nvest, for the parents or tutors, a good dstrbuton of the credts allow plannng fnancal efforts. Balanced Academc Currculum Problem (BACP) conssts n the allocaton of courses n the perods that are part of a currculum such that the prerequstes are satsfed and the credts load s balanced. The BACP belongs to the class of problems CSP (Constrant Satsfacton Problems), and ths s a decsonal optmzaton problem classfed as NP-Hard [1]. The BACP problem was ntroduced by Castro and Manzano [2] wth three test cases called BACP8, BACP10 and BACP12 ncluded n CSPLb [3] and these have been used to test models proposed by other researchers. The model proposed n [2] uses the followng nteger programmng model: Parameters m : Number of courses n : Number of perods : Number of credts of course ; =1..m : Mnmum academc load allowed per perod : Maxmum academc load allowed per perod : Mnmum amount of courses per perod : Maxmum amount of courses per perod Decson Varables x = 1 f course s assgned to perod 0 otherwse c : academc load of perod, =1,..., n c m 1 Obectve functon x 1.. n (1) Mn c Max c, (2) 1, c n Constrants If the course b has the course a as prerequste then: x a < x b c k 1., n (3) m x 1,, n (4) k 1

2 Recent works have tred to solve ths problem usng genetc algorthms and constrant propagaton [4], local search technques [5], formal methods (HyperLngo) for the nteger programmng problems [6] and multple optmzaton, usng genetc algorthm of local search [7]. All these approach have found the optmal for the three test cases ncluded n CSPLb and n some cases also for the currculums of ther unverstes. In [6] was proposed a modfed BACP problem where are consdered constrants of academc load and total of courses wthn a specfc range per perod,.e., not necessarly all perods wll have the same ranges for ther academc loads and number of courses; also add the restrcton of to locate a course n a gven perod. Ths problem was modeled as an nteger programmng problem and s reported to fnd optmum solutons usng a formal method for some of ts nstances but not for all, and the solutons for the three nstances ncluded n CSPLb. In ths paper s solved the modfed BACP usng tabu search to fnd solutons to the nstances that the formal method could not to solve 1.1 Formulaton for Model BACP Modfed. In the model of nterest proposed n [6] s consdered to modfy two constrants of the base formulaton, the frst one s to make flexble the course load per perod and the second one s to make flexble the number of courses per perod,.e., that we can place dfferent lmts on course load and number of courses for each perod. It also adds a restrcton whch allows the locaton of some of the courses n a specfc perod. Parameters Nta : Number of courses : Number of academc perods crd : Number of course credts =1..Nta mca : Mnmum academc load allowed per perod Mca : Maxmum academc load allowed per perod mna : Mnmum number of courses per perod Mna : Maxmum number of courses per perod c: Course t s desrable to locate between certan perods. mpcc: Mnmum perod of locaton of the course Mpcc: Maxmum perod of locaton of the course C : Academc load Nta C crd x (5) Decson Varables C : Academc load for the perod =1.. Cmx : Maxmum course load x = Obectve Functon f obectve Mn Cmx (6) where Cmx = Max { c 1, c 2,, c } Constrants 1 f course s assgned to perod 0 otherwse The load of the perod must be wthn the allowable range. mca C Mca 1.. (7) The number of courses of the perod must be wthn the allowable range. Nta mna x Mna If the course b has the course a as prerequste then 1 xb xar 2.. r 1 Convenent locaton for the course c 2 Tabu Search Mpc c c mpvc (8) (9) x 1 (10) The proposal soluton s based on use of heurstc n ths case tabu search. The tabu search s a method used to solve combnatoral optmzaton problems. The man dea behnd the tabu search method s that by usng a "memory" forces the method to explore new areas n the search space. That s, t can "memorze" some solutons that have been examned recently and these ponts become forbdden (taboo) to make decsons about the followng soluton. To use ths method s used the followng structure to represent the dfferent actors nvolved n the model. One element of the populaton s represented by a vector, where the poston ndcates the course and the content of each poston ndcates the perod to whch t was assgned, as shown n fgure 1.

3 Fgure 1 Element of the soluton In our case we use a short-term memory on the movements of courses by perod. To keep track of movements that have been made, and wll are prohbted, we use a two dmensonal array, of sze total number of perods (ntp), n whch s stored the perods between whch the movement took place as shown n fgure ntp-1 mn max Fgure 2 Short-term memory We can consder that a balanced currculum should have a unform dstrbuton of all the credts that make up the currculum, so the ftness functon used s the sum of the absolute error, whch s calculated usng the followng functon. Ftness( h) C k P (9) k1 Where C k s the academc load of the perod k calculated wth the formula (4) and P s the average number of credts per perod. P C / (10) 1 The ntal populaton conssts of the currculum that we want to balance, ths s a feasble soluton. Once we have the frst feasble soluton s necessary to obtan the perod wth least load and perod wth more load. And these perods are stored n the array that wll handle the short-term memory for the tabu search. Pmax = max{c k } to k=1,, ntp Pmn = mn{c k } to k=1,, ntp To generate a new possble soluton s sought the frst course n order of appearance, whch belongs to the perod wth maxmum load (Pmax). Gven the course are valdated the restrctons of prerequstes, load and preference of the perod of a course, f they are satsfed the change s made, n another way another course s chosen n the same way and redo the valdaton of the new course. In case that none of the courses reassgned to Pmax perod allow the change, the mnmum perod current s marked as nelgble and s calculate a new mnmum perod, wth whch s repeat the above process. At the tme of fndng a soluton the perods marked as nelgble are marked as elgble. The two solutons, the current and the soluton generated, are evaluated by the ftness functon (formula 10) and the best s selected for the next generaton. In each teraton s tested that the Pmax and Pmn perods are not n the tabu lst. Once that the tabu lst s flled, the counter of the tabu lst s set at 0 to begn replacng frst the old value lke a crcular lst. When s detected that a local optmum has been reached, a change n the process of generaton of a new soluton s made. Now the new soluton s generated exchangng two courses, the frst one n the perod wth more load and the second one n the perod wth less load. Havng the two courses whch wll be exchanged, are evaluated the restrctons of prerequste, load, course and perod of preference, f the exchange can be gven, the new ndvdual s generated, n other way s chosen another course of the mnmum perod and the valdaton s made newly. 3 Results The tests were carred out for the three base cases ncluded n CSPLb and the cases proposed by [5] for whch no soluton could be found. 3.1 Base Cases The base cases ncluded n CSPLb are: BACP8, BACP10 and BACP12, whose features are shown n table 1. Table 1. General features of currculums Code BACP8 BACP10 BACP12 # Total Courses # Total credts #Total Academc perod #Relaton Prerequste Mn. Courses /perod Max. Courses / perod Mn Load/ perod Max Load/ perod #Courses wth locaton Table 2 shows the result obtaned wth the proposed algorthm, n all cases the optmum s reached.

4 Table 2. Results summary Code Optmum Average Average tme Iteratons (mn.) BACP BACP BACP The academc load per perod obtaned by the algorthm s shown n table 3. Table 3. Soluton found for BACP Proposed Cases Perod Load Courses The cases not ncluded n lbrary CSPLb used to test ths algorthm are taken from [5], the frst s one for whch could not always fnd the optmal and the second s where the optmum never was found. The features of these two problems are shown n table 4. Table 4. General features of currculums Code Ic-06 Ind-06 # Total Courses # Total credts #Total Academc 9 9 perod #Relaton Prerequste Mn. Courses /perod 5 4, 4, 4, 4, 4, 4, 4, 4, 2 Max. Courses/ 8 9, 9, 9, 9, 9, 9, 9, 9, 4 perod Mn Load/ perod 20 20, 20, 20, 20, 20, 20, 20, 20, 15 Max Load/ perod 60 60, 60, 60, 60, 60, 60, 60, 60, 40 #Courses wth locaton In tables 5 and 6 s showng the courses that have preference of locaton n each of the currculums, Ic-06 and Ind-06 respectvely. Table 5. Preference of locaton Ic-06 Course Code Mnmum Perod Maxmum Perod C C C CIV CIV CIV CIV MAT MAT MAT MAT OI OI OI OI Table 6. Preference of locaton Ind-06 Course Code Mnmum Perod Maxmum Perod C C C C FHU FHU FHU IND IND IND IND IND LPCI 1 6 LPCII 1 6 OH OI OI OI OI SSC SSP Table 7 shows the results obtaned wth the algorthm; n all cases the optmum was reached.

5 Table 7. Results summary Code Optmum Average Average tme Iteratons (mn.) Ic Ind The academc load per perod obtaned by the algorthm s shown n table 8. 4 Conclusons Table 8. Soluton found for Ind-06 Perod Load Courses In ths paper we present the soluton, usng tabu search, for a modfed Balanced Academc Currculum Problem, where the load for each perod can be equal or dfferent and s allowed to have some courses n a specfc perod. In a prevous work s showed that s possble to fnd solutons wth HyperLngo for some of the nstances of the problem, but not for all of them. However by the results obtaned was proved that the use of tabu search helps to fnd solutons to the problems that could not be resolved wth the formal method. 5 References [1] J. Salazar, Programacón matemátca, Madrd: Daz de Santos, [2] C. Castro and S. Manzano, Varable and value orderng when solvng balanced academc currculum problem, Proc. of the ERCIM Workng Group on Constrants, [3] T. Lambert, C. Castro, E. Monfroy, F. Saubon, Solvng the Balanced Academc Currculum Problem wth an Hybrdzaton of Genetc Algorthm an Constrant Propagaton, In Proceedngs of ICAISC, pages , Berln: Sprnger-Verlag, [4] Luca D Gaspero, Andrea Schaerf, Hybrd Local Search Technques for the Generalzed Balanced Academc Currculum, In Proceedngs of HM, pages , Berln: Sprnger-Verlag, [5] José Antono Agular Solís, Un modelo basado en optmzacón para balancear planes de estudo en Insttucones de Educacón Superor, Phd.Tess, Puebla: UPAEP, 2008 [6] Carlos Castro, Broderck Crawford, Erc Monfroy, A Genetc Local Search Algorthm for the Multple Optmsaton of the Balanced Academc Currculum Problem, In Proceedngs of MCDM, pages , Berln: Sprnger-Verlag, 2009.