Chapter 12. Assigning Proctors to Exams with Scatter Search 1. INTRODUCTION

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1 Chapter 12 Assigig Proctors to Exams with Scatter Search RAFAEL MARTÍ, HELENA LOURENÇO AND MANUEL LAGUNA Uiversitat de Valecia, Uiversitat Pompeu Fabra ad Uiversity of Colorado Abstract: I this paper we preset a algorithm to assig proctors to exams. This NPhard problem is related to the geeralized assigmet problem with multiple objectives. The problem cosists of assigig teachig assistats to proctor fial exams at a uiversity. We formulate this problem as a iteger program (IP) with a weighted objective that combies a preferece fuctio ad a workload-fairess fuctio. We develop a scatter search procedure ad compare its outcome with solutios foud by solvig the IP model with CPLEX 6.5. Our test problems are real istaces from a Uiversity i Spai. 1. INTRODUCTION Cosider a set of teachig assistats (TAs) at a large uiversity. Each TA has a maximum umber of hours that he/she ca devote to proctor fial exams. This limit depeds o his/her cotract ad teachig load. Each fial exam requires a give umber of TAs for proctorig. The Proctor Assigmet Problem cosists of assigig the TAs to the fial exams. Sice the TAs are graduate studets, they also have fial exams ad therefore they caot proctor exams durig periods that coflict with their ow exams. The costraits ca be summarized as follows: Each exam must be proctored by a specified umber of TAs. A TA caot exceed his/her maximum umber of proctor hours. A TA caot proctor more tha oe exam at the same time. A TA caot proctor a fial exam that coflicts with oe of his/her ow. A TA should proctor the exams of the courses he/she taught. 215

2 216 Martí, Loureço ad Lagua Note that the last costrait ca be hadled before formulatig the model by simply assigig proctors to the exams of the courses they taught ad adjustig the associated iput data accordigly (e.g., reducig the total umber of proctor hours ad the exam requiremets). Teachig assistats have prefereces for some exams, which reflect their desire for proctorig o a give day or avoidig certai days. For example, some TAs would like to avoid proctorig a exam the day before oe of their exams. As a result of these prefereces, oe objective of the problem is to make assigmets that maximize a fuctio of the prefereces. Aother importat criterio that must be cosidered is to assig exams so the workload is evely distributed amog TA s. Ufair workloads are likely to geerate coflicts amog TA s ad betwee TA s ad the admiistratio. Several objective fuctios ca be formulated to measure the workloadfairess of a give assigmet. Oe possibility is to miimize the differece betwee the TA with the largest workload ad the oe with the smallest. Alteratively, the model ca seek to maximize the miimum workload associated with each TA. Sice the umber of available hours for each TA varies, the workload ca be expressed as the ratio of assiged hours to available hours. The Proctor Assigmet Problem (PAP) ca be viewed as a extesio of the well-kow Geeralized Assigmet Problem (GAP) (see Loureço ad Serra 1998; Lagua, et al. 1995; Osma 1995; ad Chu ad Beasley 1997), by allowig more tha oe aget to be assiged to a task ad itroducig side costraits. The side costraits model aspects of the problem that are ot part of the GAP, such as the eed for assigig multiple proctors to a exam ad for avoidig the assigmet of the same TA to multiple exams that are held at the same time period. Moreover, istead of a sigle objective fuctio that maximizes the total profit i the GAP, the PAP cosiders two objective fuctios to simultaeously maximize prefereces ad fairess. Therefore, the PAP ca be formulated as multi-objective iteger program. We propose a heuristic procedure based o the scatter search methodology to fid solutios to the PAP. The procedure will be embedded i a user-friedly system to help the plaig of fial exams i the School of Ecoomics of the Uiversity Pompeu Fabra at Barceloa (Spai). 2. PROBLEM FORMULATION I order to simplify the problem ad mimic the maual method curretly practiced at the Uiversity Pompeu Fabra of Barceloa, we split a exam

3 Assigig Proctors to Exams with Scatter Search 217 day ito two periods (8:00 AM - 2:00 PM ad 2:00 PM - 9:00 PM). The if d is the total umber of exam days, k = 2d is the total umber of periods. The followig assumptios are made: A exam starts ad eds i the same day. A exam ca be scheduled over oe or two periods i a give day. TA s express their prefereces for give periods. I order to stadardize the TA prefereces, we traslate the prefereces for periods to prefereces for exams. That is, if a TA expresses a strog preferece for period 1 (8:00 AM - 2:00 PM i the first day of fial exams), the we cosider that the TA has a strog preferece for all exams scheduled durig this period. If a exam is scheduled over two periods (e.g., a exam that starts at oo ad fiishes at 3:00 PM o the first day of fial exams), the we use the lower value betwee the preferece for period 1 ad 2, as expressed by each TA. Let J be the set of m exams (j = 1,..., m) ad I the set of TA s (i = 1,..., ). The, the followig otatio is used to represet the relevat data i our problem: a i = maximum umber of available hours for TA i. b j = umber of hours associated with exam j. t j = umber of TA s required for exam j. c = preferece of TA i for the exam j. P i = the set of exams that overlap with ay TA i s exams. T k = the set of exams scheduled i period k. d = umber of exam days. Also, we defie the set of biary variables x, such that x = 1 if TA i is assiged to exam j; ad x = 0 otherwise. Usig these defiitios, the PAP ca be formulated as follows: Max f ( x) = i= 1 j= 1 c x Mi g ( x) = y (2) (1) Subject to m b j x j=1 a i i = 1,, (3)

4 218 Martí, Loureço ad Lagua m j= 1 b j x ai y 0 i = 1,, (4) x = i=1 i T k t j j = 1,, m (5) x 1 i = 1,, ; k = 1,, 2d (6) 0,1 x = i = 1,, ; j = 1,, m (7) x = 0 j Pi (8) y 0 (9) Equatio (1) ad (2) represet the sum of the prefereces ad the miimum TA utilizatio, respectively. These objectives are to be maximized. The utilizatio is the fractio of the available hours that a TA is assiged to proctor exams. Costrait (3) limits the umber of assiged hours to ot exceed the available hours for each TA. Costrait (4) calculates the miimum TA utilizatio. Costrait (5) guaratees that each exam has the required umber of TA s. Costrait (6) guaratees that each TA ca proctor at most oe exam i the same period. Costrait (7) eforces the biary restrictios o the decisio variables. Costrait (8) elimiates the assigmets that create a coflict with the exams that proctors have. Fially, costrait (9) eforces the oegativity restrictio o the y- variable. We combie the objective fuctios to create a sigle, weighted, fuctio of the followig form: h ( x) f ( x) + αg ( x) = (10) where ( ) f x = m c i= 1 j= 1 max m j= 1 c t j ( j) x (11) ad c ( j ) max( 1, c ) max =. (12) i

5 Assigig Proctors to Exams with Scatter Search 219 Note that sice the miimum preferece value is zero, the both f ( x) ad g ( x) are bouded betwee 0 ad 1. The value of α 0 is used to show the preferece of the decisio maker towards either elemet of the combied objective fuctio. If α > 1, the the decisio maker prefers assigmets with a more uiform distributio of the workload. If α < 1, the the decisio maker prefers assigmets that maximize the total (ormalized) preferece value. 3. SCATTER SEARCH APPROACH Scatter search, from the stadpoit of metaheuristic classificatio, may be viewed as a evolutioary (or also called populatio-based) algorithm that costructs solutios by combiig others. It derives its foudatios from strategies origially proposed for combiig decisio rules ad costraits (i the cotext of iteger programmig). The goal of this methodology is to eable the implemetatio of solutio procedures that ca derive ew solutios from combied elemets i order to yield better solutios tha those procedures that base their combiatios oly o a set of origial elemets. As described i tutorial articles (Glover 1998 ad Lagua 1999) ad other implemetatios based o this framework (Campos et al. 1998), the methodology icludes the followig basic elemets: Geerate a populatio P. Extract a referece set R. Combie elemets of R ad maitai ad update R. Scatter search fids improves solutios by combiig solutios i R. This set, kow as the referece set, cosists of solutios of high quality that are also diverse. The overall proposed procedure, based o the scattersearch elemets listed above, follows: Preprocessig: Exams with a small umber of studets are proctor by the course s professor. TA s are assiged to the exams correspodig to the courses they taught. Exam ad TA data are updated to reflect these assigmets. Geerate a populatio P: Apply the diversificatio geerator to geerate diverse solutios.

6 220 Martí, Loureço ad Lagua Costruct the referece set R: Add to R the best r 1 solutios i P. Add also r 2 diverse solutios from P to costruct R with R = r 1 + r 2 solutios. Maitai ad update the referece ret R: Apply the subset geeratio method (Glover 1998) to combie solutios from R. Update R, addig solutios that improve the quality of the worst i the set. We ow describe the implemetatio details of the mai elemets of the procedure, as adapted i the cotext of the exam proctor assigmet. 3.1 Diversificatio Geeratio Method A iitial populatio of solutios is costructed by meas of a diversificatio geerator. The geerator that we implemeted is based o the otio of costructig solutios employig modified frequecies. The goal of the method is to geerate good-quality diverse solutios. The geerator uses the followig frequecy fuctio: f = (13) x x P This frequecy value is used to bias the potetial assigmet of TA i to exam j durig subsequet costructios of solutios, ad therefore to iduce diversity i the ew solutios with respect to the solutios already i P. The attractiveess of assigig TA to a exam is give by his/her preferece. The preferece of TA i for exam j (c ) is a value i the rage [0,5]. The preferece value is computed as the differece betwee the period i which exam j is scheduled ad the period of the closest exam that TA i must take. Differeces of more tha 5 periods are adjusted back to 5. The period just before oe for which a TA has a exam is assiged a preferece of 0. We modify the value of c to reflect previous assigmets of TA i to exam j, as follows: c' max β c (14) i,j = c f maxi,j f It should be oted that c' is a adaptive fuctio, sice its value depeds o attributes of the uassiged elemets ad a fuctio of previous assigmets. The value of β is dyamically modified to ecourage

7 Assigig Proctors to Exams with Scatter Search 221 additioal diversificatio. If the geerator costructs the same solutio more tha oce, the β -value is icreased. I our implemetatio, we start with β = 0.4 ad icrease its value by 0.1 every time the geerator repeats a costructio. Figure 1 summarizes the diversificatio geeratio method. The method geerates PopSize solutios usig the updated c' values. TA s are assiged to exams i order to maximize the modified preferece values. The procedure stops whe PopSize solutios are geerated. Iitializatio Solutios = 0; f =0 for all i,j While (Solutios < PopSize ) For k=1 to 2d Order the exams i period k by decreasig umber of required TA s. For each exam j i period k Costruct the list of TA s that ca proctor j. c Order the list accordig to Assig the first t j TA s to proctor exam j. Update TA ad exam iformatio. Add the solutio to the populatio. Make Solutios = Solutios + 1; Update the correspodig f. Figure 1. Diversificatio geerator 3.2 Updatig ad Maitaiig the Referece Set The referece set R is a subset of the populatio set P that cosists of x, x, betwee high quality ad diverse solutios. A distace fuctio ( ) two solutios x ad x, is used to measure the diversity of the solutios i the referece set.

8 222 Martí, Loureço ad Lagua ( x x ) = m, x x (15) i= 1 j= 1 Note that a large distace betwee two solutios does ot traslate ito a large differece betwee their correspodig objective fuctio values. This is why diversity of the solutios i R caot be measure with referece to the objective fuctio values oly. The referece set R with R = b is costructed as follows. Order the solutios i P by decreasig value of the objective fuctio. Select the first b/2 solutios ad add them to R. For each solutio x i P-R, calculate the distace to all the elemets i R. Let the miimum distace mi ( x) from a solutio x i P-R to all solutios x i R be defied as: ( x) = mi ( x, x ) mi (16) x R Select the solutio * x with the maximum distace ( x) of all x i P- * R. Add x to R, util R = b. I our experimets, we use b = 20, because previous studies (Campos, et al ad 1999) have show the merit of such choice. 3.3 Solutio Combiatio Method Scatter search geerates ew solutios by combiig those i the referece set. Specifically, a combiatio method is applied to subsets of solutios i the referece set. A ewly geerated solutio is compared with the worst solutio i R. The ew solutio replaces the worst solutio if the objective value of the ew solutio is better tha the objective value of the worst solutio i R. The solutio combiatio procedure seeks to geerate subsets X of R that have useful properties, while avoidig the duplicatio of subsets previously geerated. The approach for doig this is orgaized to geerate four differet collectios of subsets of R, which Glover (1998) refers to as SubSetType 1, 2, 3 ad 4: mi SubsetType 1: SubsetType 2: all 2-elemet subsets. 3-elemet subsets derived from the 2-elemet subsets by augmetig each 2-elemet subset to iclude the best solutio ot i this subset.

9 Assigig Proctors to Exams with Scatter Search 223 SubsetType 3: 4-elemet subsets derived from the 3-elemet subsets by augmetig each 3-elemet subset to iclude the best solutios ot i this subset. SubsetType 4: the subsets cosistig of the best i elemets, for i = 5 to b. A cetral cosideratio of this desig is that R itself might ot be static, because it might be chagig as ew solutios are added to replace old oes (whe these ew solutios qualify to be amog the curret b best solutios foud). The solutio combiatio method is applied to each subset. This is based o a votig system, where each solutio votes for specific assigmets of TA s to exams. The resultig costructio may be ifeasible with respect to costraits (3) ad (6), i which case, a repair mechaism is applied. The votig scheme is summarized i Figure 2. For each exam j Fid the assigmet x with the largest vote. Assig the exam j to TA i. Figure 2. Votig mechaism The vote associated with the assigmets x j is a measure of the merit of assigig TA s to exam j. The vote is therefore defied to take ito accout the preferece of the TA s for some exams. I additio, the vote pealizes assigmets that result i violatios of oe or more of the problem costraits. The process cosists of m steps (where m is the umber of exams). At each step j, a solutio x votes for its colum x j, that cosists of all the TA assigmets associated with exam j. The vote of solutio x at step j is calculated as follows: V j ( x) = i= 1 c x ϕ max 0 γ max 0, k:j Tk j 1, l = 1 b y + b x il il l T k,l j 1 l y j + x a i 1 (17)

10 224 Martí, Loureço ad Lagua The first term of the vote calculatio adds the prefereces of the TA s assiged to exam j i solutio x. The secod term calculates the umber of hours assiged to each TA i the solutio that is beig costructed (represeted by the y-variable). The, it adds the umber of hours for the curret exam j ad the subtracts the available hours for the TA. If the hours already assiged plus the hours related to the curret exam exceed the available umber of hours, the the excess hours are multiplied by a pealty factor ϕ. I other words, the secod term i equatio (17) pealizes violatios of costrait (3). I a similar way, the third term of equatio (1) pealizes the violatio of costrait (6), by multiplyig the umber of times a TA is assiged to proctor differet exams i the same time period by the costat γ. Whe the combiatio method results i a solutio that violates either costrait (3) or costrait (6), the procedure i Figure 3 is executed i a attempt to repair the ewly geerated solutio. For each period k For each exam j scheduled i period k If ay TA assiged to j is also assiged to aother exam i period k or his/her capacity has bee exceeded Fid the ext available TA that ca be feasibly assiged to exam j. Stop if o such TA ca be foud. Figure 3. Solutio repair procedure Although the procedure i Figure 3 may fail to repair a solutio, our experiece has bee that the method seldom fails.

11 Assigig Proctors to Exams with Scatter Search COMPUTATIONAL EXPERIMENTS The data used for these experimets correspod to real istaces of the proctor assigmet problem at the Uiversitat Pompeu Fabra i Barceloa (Spai). Presetly, proctors are maually assiged to exams, followig simple rules that eforce the costraits i the problem. We compare our results with assigmets that were geerated maually ad also with assigmets foud solvig the mixed-iteger programmig formulatio with Cplex 6.5 (some of which are optimal). Our test set cosists of 11 problems. The results are summarized i Table 1. The first three colums i this table show the problem umber, the umber of TA s ad the umber of exams, respectively. The table tha shows the objective fuctio value correspodig to the scatter search solutio ad the solutio foud with Cplex. The asterisk i the Cplex colum idicates that the solutio was cofirmed to be optimal. Table 1 shows that scatter search solutios are iferior to those foud by Cplex i the set of problems used for testig ad the objective fuctio chose to measure performace. However, a advatage of scatter search is that the referece set cotais a umber of high-quality solutios from which the decisio-maker could choose the oe to implemet. The maximum stadard deviatio of the objective fuctio value for solutios i the fial referece set was for all problem istaces. This idicates that practically all of the solutios i the fial referece set have the same quality with respect to the objective fuctio value. Sice the objective fuctio is a mathematical represetatio of some subjective measure of performace associated with a give assigmet, the ability to choose amog solutios that have similar objective fuctio values is a importat feature of a solutio procedure to be embedded i a decisio support system desiged for this maagerial situatio.

12 226 Martí, Loureço ad Lagua Table 1. Summary of results Problem TA s Exams Scatter Search Cplex (* = optimal) * * * * The scatter search procedure was coded i C ad ru o a Petium II computer at 350 MHz. The computatioal times are very reasoable, with a average of secods ad a maximum of secods. Cplex 6.5 was also ru o a Petium II machie at 350 MHz. The depth-first search strategy was used with a time limit of oe hour. 5. CONCLUSIONS Oe of the most iterestig features of the proctor assigmet problem is its multi-objective ature. I the curret developmet, we have formulated the problem as a mixed-iteger program with a sigle objective ad implemeted a scatter search solutio procedure. We aticipate that this procedure ca be the basis for oe that directly exploits the multi-objective characteristics of this problem. This geeralizatio ca be achieved by costructig ad updatig a referece set R cosistig of o-domiated solutios. A solutio x is domiated if it exits aother solutio x i R such that f ( x ) > f ( x) ad g ( x ) < g ( x). I other words, sice the objectives of the x g x, a solutio x is problem are to maximize f ( ) ad miimize ( ) domiated whe there is a solutio i x that is better tha x with respect to both objectives. This defiitio would create a dyamic update of R, where the cardiality of the set would vary with the set of o-domiated solutios. The implemetatio of a procedure that uses these defiitios ad is based o the mechaisms developed i this paper will be the topic of a future project.

13 Assigig Proctors to Exams with Scatter Search 227 REFERENCES Awad, R. ad J. Chieck (1998) Proctor Assigmet at Carleto Uiversity, Iterfaces, vol. 28, o. 2, pp Campos, V., F. Glover, M. Lagua ad R. Martí (1999) A Experimetal Evaluatio of a Scatter Search for the Liear Orderig Problem, Uiversity of Colorado at Boulder. Campos, V., M. Lagua ad R. Martí (1998) Scatter Search for the Liear Orderig Problem, to appear i New Methods i Optimisatio, D. Core, M. Dorigo ad F. Glover (Eds.), McGraw-Hill. Chu, P. C. ad J. E. Beasley (1997) A Geetic Algorithm for the Geeralized Assigmet Problem, Computers ad Operatios Research, vol. 24, pp Glover, F. (1998) A Template for Scatter Search ad Path Relikig, i Artificial Evolutio, Lecture Notes i Computer Sciece 1363, J.-K. Hao, E. Lutto, E. Roald, M. Schoeauer ad D. Syers (Eds.), Spriger-Verlag, pp Lagua, M., J. P. Kelly, J. L. Gozález Velarde ad F. Glover (1995) Tabu Search for the Multilevel Geeralized Assigmet Problem, Europea Joural of Operatioal Research, vol. 82, pp Loureco HR ad Serra D, 1998, Adaptive approach heuristics for the geeralized assigmet problem', DEE-UPF, Ecoomic Workig paper Series No. 288, May. Osma, I. H. (1995) Heuristics for the Geeralized Assigmet Problem: Simulated Aealig ad Tabu Search Approaches, OR Spektrum, vol. 17, pp