# Assigning judges to competitions of several rounds using Tabu search

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<ul><li><p>e, judges have to be selected to evaluate the performance of the competitors and to identify a winner.ered when assigning judges to the competitions. These requirements include the ofcial rules of theand thny ruleis difc</p><p>e assigCanadd to ot</p><p>rience, and the set of new judges participating for the rst time. Note that all lead judges are experienced but the converse is false. Inaddition, 6 different elds of expertise for the judges are considered, and each judge indicates at least one of these expertises. Also, alljudges are uent in English, but only some of them are also uent in French. Finally, each judge indicates the rounds for which he isavailable.</p><p>Following the round-robin tournament, the best teams move to the nals in the second part of the competition.</p><p> Corresponding author. Tel.: +1 514 343 5687; fax: +1 514 343 5834.</p><p>European Journal of Operational Research 210 (2011) 694705</p><p>Contents lists available at ScienceDirect</p><p>European Journal of Operational ResearchE-mail addresses: amina.lamghari@mail.mcgill.ca (A. Lamghari), ferland@iro.umontreal.ca (J.A. Ferland).Every year, the John Molson School of Business organizes the John Molson International Case Competition involving 30 teams of busi-ness students coming from top international universities. This set of teams is partitioned into 5 groups, each including 6 teams. The rstpart of the competition consists of a round-robin tournament including 5 rounds where each team competes against each of the other 5teams of its group. Thus, each round includes 15 individual competitions, giving a total of 75 individual competitions that take place overthe rst part of the competition.</p><p>In each round, all teams have the same specied business case to analyze, to evaluate, and to propose solutions for it. After a 3 hoursperiod to prepare their presentation, each team of an individual competition makes an oral presentation in front of a panel of judges. Priorto the competition, teams specify the language (English or French) that they will use during their presentations.</p><p>More than 200 university professors and/or senior business executives representing various rms are usually available for judgingthe presentations. According to their past experience and other considerations, the judges are divided into 3 sets: the set of leadjudges having the skills to chair the jury in a given individual competition, the set of experienced judges having previous judging expe-Whenever competitions take placSeveral requirements must be considcompetition that cannot be violated,most of the competitions involve masuming task for the staff involved. Thgenerating the assignments.</p><p>In this paper, we analyze the judgat Concordia University in Montreal (should nevertheless be easily adapte0377-2217/$ - see front matter 2010 Elsevier B.V. Adoi:10.1016/j.ejor.2010.10.034e objectives of the organizing committee that should be satised as much as possible. Becauses and several objectives, manually generating the assignments is often a difcult and time-con-ulty motivates the interest of the scientic community in developing automated procedures for</p><p>nment problem for the John Molson International Case Competition that takes place every yeara) for more than 25 years. Even if the solution approach is introduced for this specic context, ither contexts by making proper minor adjustments to deal with slightly different specic rules.Innovative Applications of O.R.</p><p>Assigning judges to competitions of several rounds using Tabu search</p><p>Amina Lamghari a, Jacques A. Ferland b,aCOSMO Stochastic Mine Planning Laboratory, McGill University, Department of Mining and Materials Engineering, FDA Building,3450 University Street, Montreal, Quebec, Canada H3A 2A7bDpartement dInformatique et de Recherche Oprationnelle, Universit de Montral, C.P. 6128, Succursale Centre-Ville, Montral, Qubec, Canada H3C 3J7</p><p>a r t i c l e i n f o</p><p>Article history:Received 21 January 2010Accepted 27 October 2010Available online 9 November 2010</p><p>Keywords:Tabu searchSchedulingAssignment</p><p>a b s t r a c t</p><p>The judge assignment problem consists in nding an assignment satisfying the competition rules (hardconstraints) and meeting, as much as possible, the competition organizers objectives (soft constraints).In this paper, various specic real-world constraints found in organizing academic competitions are han-dled. We tackle the corresponding problem with a metaheuristic approach based on Tabu search. Thenumerical results indicate that very good solutions can be generated in reasonable computational times.</p><p> 2010 Elsevier B.V. All rights reserved.</p><p>1. Introduction</p><p>journal homepage: www.elsevier .com/locate /e jorll rights reserved.</p></li><li><p>The teams schedules are established by the organizing committee. In this paper, we are interested in generating the judge assignmentsfor the rst part of the competition. These assignments should fulll specic rules which are divided into 2 categories:</p><p>Hard rules or constraints that must be satised: Since all the individual competitions of a specic round take place simultaneously, an available judge can be assigned to at most one</p><p>individual competition of the round. 3 or 5 judges must be assigned to each individual competition. At least one of the judges belongs to the set of lead judges. At least one of the judges, different from the lead judge, belongs to the set of experienced judges. A judge cannot be assigned to an individual competition involving a team coming from a University where he received his degree or</p><p>where he is a faculty member. A judge cannot be assigned to an individual competition involving a team which he does not wish to evaluate. If a team in an individual competition is presenting in French, then the judges assigned to this individual competition must also be</p><p>uent in French.Soft rules or constraints (or objectives) to be satised as much as possible: Balance constraints: in each individual competition, the number of experienced judges assigned should be equal to the number of new</p><p>judges. Afliation constraints: if several judges assigned to an individual competition are coming from rms, then they should come from</p><p>different ones. Diversity constraints: the expertises of the judges assigned to an individual competition should cover as many of the 6 elds of exper-</p><p>tise as possible. Coupling constraints: two judges should not be assigned together more than once during all rounds. Sequence constraints: during the different rounds, a judge should not be assigned to different individual competitions involving the</p><p>same team. Number constraints: the number of individual competitions having 5 judges assigned should be maximized.</p><p>In previous works (Lamghari and Ferland, 2005, 2007, 2010), we consider a simplied version of the problem and propose heuristictechniques and metaheuristic methods related to Tabu search (Glover and Laguna, 1998; Hansen, 1986) to solve it. In the simplied ver-sion, the following assumptions are made:</p><p> There are only two sets of judges: the set of lead judges and the set of other judges. Therefore, the fourth hard constraint and the balanceconstraints are not considered.</p><p> The judges preferences are not taken into account when assigning them to individual competitions, and thus the sixth hard constraint isnot considered.</p><p> We assume that the competition language is English, hence the last hard constraint is not taken into account. All judges are university professors. Accordingly, afliation constraints need not to be considered. Each judge has only one expertise. The diversity constraints are thus reduced to the requirement that the expertises of the judgesassigned should be as different as possible.</p><p> We were solving the problem involving only one round, and thus coupling and sequence constraints are not considered.</p><p>The metaheuristic methods in Lamghari and Ferland (2010) have proved very efcient for the simplied version of the problem involv-ing one round. Thus, it is worthwhile examining their efciency for the more difcult problem involving several rounds and including morerealistic constraints. Furthermore, to solve the problem associated with several rounds, we consider two different alternatives. The rst oneis a global approach dealing simultaneously with the different rounds. It is obtained by a straightforward extension of the Tabu searchmethod in Lamghari and Ferland (2010). In the second alternative, we develop a decomposition approach separating the problem into aseries of sub-problems, each associated with one round. The sub-problems are considered sequentially and the solution of each sub-prob-lem is improved by xing the solutions of the other sub-problems to their current values.</p><p>We provide numerical results allowing to evaluate and compare the performance of the proposed solution approaches. These resultsindicate that the decomposition approach generates better solutions requiring smaller solution time.</p><p>Several papers appeared in the literature reporting the study of the judge assignment problem in the context of sport competitions. Theassignment rules and the objectives differ with the specic contexts. For instance, some constraints are related to the number of judgesrequired, to their level of experience, and to the sequence of assignments of a given judge. The objective function might be to minimizethe total traveled distance of the judges, or to balance their workload, or to minimize a weighted sum of violations of the soft constraints,for instance. These applications induce difcult combinatorial optimization problems usually solved with heuristic or metaheuristic meth-ods. A three-phase approach based on a constructive heuristic, a repair heuristic, and an iterated local search improvement heuristic wasproposed by Duarte et al. (2006, 2007b). This approach was extended in Duarte et al. (2007a) by using an exact algorithm in the third phase.Yavuz et al. (2008) develop a two-phase approach including a constructive heuristic and a local search procedure. Evans et al. (1984) use aminimum cost ow formulation whilst Ordonez and Knowles (1998) use a constraint satisfaction formulation. Also, Evans (1988), Wright(1991), and Farmer et al. (2007) develop decision support systems for baseball, cricket, and tennis leagues, respectively. A more detailedreview of these problems is given by Wright (2009), and Kendall et al. (2010).</p><p>The remainder of the paper is organized as follows: In Section 2 a mathematical formulation of the problem is introduced. Section 3includes the decomposition approach. Section 4 provides a description of the mathematical formulation of the sub-problems. Themetaheuristic method used to solve them is summarized in Sections 5 and 6. The global approach is briey described in Section 7.</p><p>A. Lamghari, J.A. Ferland / European Journal of Operational Research 210 (2011) 694705 695Computational results on randomly generated instances are reported and discussed in Section 8. Finally, some conclusions are drawn inSection 9.</p></li><li><p> P: p: M: j: i t: t</p><p> K: k:</p><p> S =</p><p> A =</p><p>Note t</p><p> E =</p><p> R =ric 0 otherwise: The variables xijp represent the assignment of judge i to individual competition j of round p:</p><p>xijp 1 if judge i is assigned to individual competition j of round p;0 otherwise:</p><p> The variables y3jp and y5jp indicate the number of judges assigned to individual competition j of round p:</p><p>y3jp 1 if 3 judges are assigned to individual competition j of round p;0 otherwise;</p><p>y5jp 1 if 5 judges are assigned to individual competition j of round p;0 otherwise:</p><p> For modeling the balance constraints, we use the variables d1jp to denote the surplus of experienced judges over new judges, and d1jp todenote its shortage in individual competition j of round p.</p><p> Also, with each of the other soft rules, we associate a deviation variable taking the value 0 if the corresponding rule is satised, or it isequal to the number of violations of this rule, otherwise. We denote by d2cjp; d</p><p>3kjp; d</p><p>4ir , and d</p><p>5it the deviation variables associated with the</p><p>afliation, diversity, coupling, and sequence constraints, respectively. Finally, f1, . . . , f6 represent the weights associated with the soft rules.Thejudge[ric] where</p><p>1 if judge i comes from firm c;eik 0 otherwise:[eik] where</p><p>1 if judge i has expertise k; li 0 otherwise:</p><p> v i 1 if judge i is an experienced judge but not a lead judge0 otherwise:</p><p> D = [dip] where</p><p>dip 1 if judge i is available for round p;0 otherwise:</p><p>hat judge i is admissible for individual competition j of round p if the last three hard rules are satised.</p><p>1 if judge i is a lead judgeaijp 0 otherwise:[aijp] where</p><p>1 if judge i is admissible for individual competition j of round p;stjp 0 otherwise:[stjp] where</p><p>1 if team t competes in individual competition j of round p; C: the number of rms. c: rm index, c = 1, . . . ,C.the number of elds of expertise.eld of expertise index, k = 1, . . . ,K. N: the total number of judges. i, r: judge index, i, r = 1, . . . ,N.the number of rounds.round index, p = 1, . . . ,P.the number of individual competitions per round.ndividual competition index, j = 1, . . . ,M.eam index, t = 1, . . . ,2M.2. Mathematical formulation</p><p>Referring to the assignment rules described in the previous section, the problem can be formulated as an integer goal program. The hardrules are formulated as model constraints. The remaining rules are modeled as soft constraints with different importance weights, and theobjective function to be minimized is the weighted sum of the violations of the soft constraints.</p><p>We use the following notation to formulate the problem:</p><p>696 A. Lamghari, J.A. Ferland / European Journal of Operational Research 210 (2011) 694705proposed model is summarized in Fig. 1. The constraints (2)(6) are related to the hard rules. The constraints (2) indicate that acan be assigned to an individual competition of a given round only if he is available for that round. Furthermore, he cannot be</p></li><li><p>A. Lamghari, J.A. Ferland / European Journal of Operational Research 210 (2011) 694705 697assigned to more than one individual competition per round. The constraints (3) allow only admissible judges to be assigned to anindividual competition. At least one lead judge and another experienced judge, different from the lead judge, are assigned to each individualcompetition of each round according to the constraints (4). The constraints (5) and (6) guarantee that 3 or 5 judges are assigned to eachindividual competition of each round.</p><p>The constraints (7)(11) are related to the soft rules. The constraints (7) specify that, if we are not counting the lead judge, the number ofexperienced and new judges assigned to each individual competition should be equal, otherwise, one of the deviation variables d1jp or d</p><p>1jp</p><p>takes a positive value and the penalty f1 d1jp d1jp</p><p> is added to the objective function. Each rm c is represented at most once in each indi-</p><p>vidual competition according to the constraints (8), otherwise, the penalty f2d2cjp is incurred. The constraints (9) stipulate that each eld of</p><p>expertise k is...</p></li></ul>