Download - Assigning judges to competitions of several rounds using Tabu search

European Journal of Operational Research 210 (2011) 694–705

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European Journal of Operational Research

journal homepage: www.elsevier .com/locate /e jor

Innovative Applications of O.R.

Assigning judges to competitions of several rounds using Tabu search

Amina Lamghari a, Jacques A. Ferland b,⇑a COSMO – Stochastic Mine Planning Laboratory, McGill University, Department of Mining and Materials Engineering, FDA Building,3450 University Street, Montreal, Quebec, Canada H3A 2A7b Département d’Informatique et de Recherche Opérationnelle, Université de Montréal, C.P. 6128, Succursale Centre-Ville, Montréal, Québec, Canada H3C 3J7

a r t i c l e i n f o a b s t r a c t

Article history:Received 21 January 2010Accepted 27 October 2010Available online 9 November 2010

Keywords:Tabu searchSchedulingAssignment

0377-2217/$ - see front matter � 2010 Elsevier B.V. Adoi:10.1016/j.ejor.2010.10.034

⇑ Corresponding author. Tel.: +1 514 343 5687; faxE-mail addresses: [email protected] (

The judge assignment problem consists in finding an assignment satisfying the competition rules (hardconstraints) and meeting, as much as possible, the competition organizers objectives (soft constraints).In this paper, various specific 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.

� 2010 Elsevier B.V. All rights reserved.

1. Introduction

Whenever competitions take place, judges have to be selected to evaluate the performance of the competitors and to identify a winner.Several requirements must be considered when assigning judges to the competitions. These requirements include the official rules of thecompetition that cannot be violated, and the objectives of the organizing committee that should be satisfied as much as possible. Becausemost of the competitions involve many rules and several objectives, manually generating the assignments is often a difficult and time-con-suming task for the staff involved. This difficulty motivates the interest of the scientific community in developing automated procedures forgenerating the assignments.

In this paper, we analyze the judge assignment problem for the John Molson International Case Competition that takes place every yearat Concordia University in Montreal (Canada) for more than 25 years. Even if the solution approach is introduced for this specific context, itshould nevertheless be easily adapted to other contexts by making proper minor adjustments to deal with slightly different specific rules.

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 firstpart 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 first part of the competition.

In each round, all teams have the same specified 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.

More than 200 university professors and/or senior business executives representing various firms 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-rience, and the set of new judges participating for the first time. Note that all lead judges are experienced but the converse is false. Inaddition, 6 different fields of expertise for the judges are considered, and each judge indicates at least one of these expertises. Also, alljudges are fluent in English, but only some of them are also fluent in French. Finally, each judge indicates the rounds for which he isavailable.

Following the round-robin tournament, the best teams move to the finals in the second part of the competition.

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: +1 514 343 5834.A. Lamghari), [email protected] (J.A. Ferland).

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A. Lamghari, J.A. Ferland / European Journal of Operational Research 210 (2011) 694–705 695

The teams’ schedules are established by the organizing committee. In this paper, we are interested in generating the judge assignmentsfor the first part of the competition. These assignments should fulfill specific rules which are divided into 2 categories:

Hard rules or constraints that must be satisfied:� Since all the individual competitions of a specific round take place simultaneously, an available judge can be assigned to at most one

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

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

fluent in French.Soft rules or constraints (or objectives) to be satisfied as much as possible:� Balance constraints: in each individual competition, the number of experienced judges assigned should be equal to the number of new

judges.� Affiliation constraints: if several judges assigned to an individual competition are coming from firms, then they should come from

different ones.� Diversity constraints: the expertises of the judges assigned to an individual competition should cover as many of the 6 fields of exper-

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

same team.� Number constraints: the number of individual competitions having 5 judges assigned should be maximized.

In previous works (Lamghari and Ferland, 2005, 2007, 2010), we consider a simplified 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 simplified ver-sion, the following assumptions are made:

� 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.� The judges’ preferences are not taken into account when assigning them to individual competitions, and thus the sixth hard constraint is

not considered.� We assume that the competition language is English, hence the last hard constraint is not taken into account.� All judges are university professors. Accordingly, affiliation 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 judges

assigned should be as different as possible.� We were solving the problem involving only one round, and thus coupling and sequence constraints are not considered.

The metaheuristic methods in Lamghari and Ferland (2010) have proved very efficient for the simplified version of the problem involv-ing one round. Thus, it is worthwhile examining their efficiency for the more difficult problem involving several rounds and including morerealistic constraints. Furthermore, to solve the problem associated with several rounds, we consider two different alternatives. The first 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 fixing the solutions of the other sub-problems to their current values.

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.

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 specific 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 difficult 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 flow 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).

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 briefly described in Section 7.Computational results on randomly generated instances are reported and discussed in Section 8. Finally, some conclusions are drawn inSection 9.

696 A. Lamghari, J.A. Ferland / European Journal of Operational Research 210 (2011) 694–705

2. Mathematical formulation

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.

We use the following notation to formulate the problem:

� P: the number of rounds.� p: round index, p = 1, . . . ,P.� M: the number of individual competitions per round.� j: individual competition index, j = 1, . . . ,M.� t: team index, t = 1, . . . ,2M.� N: the total number of judges.� i, r: judge index, i, r = 1, . . . ,N.� K: the number of fields of expertise.� k: field of expertise index, k = 1, . . . ,K.� C: the number of firms.� c: firm index, c = 1, . . . ,C.� S = [stjp] where

stjp ¼1 if team t competes in individual competition j of round p;0 otherwise:

�

� A = [aijp] where

aijp ¼1 if judge i is admissible for individual competition j of round p;

0 otherwise:

�
Note that judge i is admissible for individual competition j of round p if the last three hard rules are satisfied.
� li ¼1 if judge i is a lead judge0 otherwise:

�

� v i ¼1 if judge i is an experienced judge but not a lead judge0 otherwise:

�

� D = [dip] where

dip ¼1 if judge i is available for round p;

0 otherwise:

�

� E = [eik] where

eik ¼1 if judge i has expertise k;

0 otherwise:

�

� R = [ric] where

ric ¼1 if judge i comes from firm c;

0 otherwise:

�

� The variables xijp represent the assignment of judge i to individual competition j of round p:

xijp ¼1 if judge i is assigned to individual competition j of round p;

0 otherwise:

�

� The variables y3jp and y5

jp indicate the number of judges assigned to individual competition j of round p:

y3jp ¼

1 if 3 judges are assigned to individual competition j of round p;0 otherwise;

�

y5jp ¼

1 if 5 judges are assigned to individual competition j of round p;

0 otherwise:

�

� For modeling the balance constraints, we use the variables d1þjp to denote the surplus of experienced judges over new judges, and d1�

jp todenote its shortage in individual competition j of round p.� Also, with each of the other soft rules, we associate a deviation variable taking the value 0 if the corresponding rule is satisfied, or it is

equal to the number of violations of this rule, otherwise. We denote by d2cjp; d

3kjp; d

4ir , and d5

it the deviation variables associated with theaffiliation, diversity, coupling, and sequence constraints, respectively.� Finally, f1, . . . , f6 represent the weights associated with the soft rules.

The proposed model is summarized in Fig. 1. The constraints (2)–(6) are related to the hard rules. The constraints (2) indicate that ajudge can be assigned to an individual competition of a given round only if he is available for that round. Furthermore, he cannot be

Fig. 1. Mathematical model.


assigned 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.

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 d1þ

jp or d1�jp

takes a positive value and the penalty f1 d1þjp þ d1�

jp

� �is added to the objective function. Each firm c is represented at most once in each indi-

vidual competition according to the constraints (8), otherwise, the penalty f2d2cjp is incurred. The constraints (9) stipulate that each field of

expertise k is covered at least once in each individual competition, otherwise, the penalty f3d3kjp is added to the objective function. The con-

straints (10) and (11) are related to the rules interconnecting the rounds. The values of the variables d4ir and d5

it are evaluated as the numberof times that the same pair of judges (i,r) is assigned more than once and the number of times that a judge i is assigned to individual com-petitions involving a team t more than once, respectively.

The objective function (1) to be minimized is equal to the weighted sum of the violations of the soft rules. It also includes the number ofindividual competitions with 3 judges assigned being penalized by a large factor f6 (which corresponds to maximizing the number of thosehaving 5 judges assigned).

In order to compare numerically the proposed solution methods with an exact method to solve the problem, we linearize the quadraticconstraints (10) to obtain a corresponding integer linear program. Note that three different linearizations are proposed in Lamghari (2008),however, the numerical results in Section 8 indicate that it is unlikely appropriate to use an exact approach in practice. Firstly, such anapproach could be used to solve only instances of small size. Secondly, the use of an exact approach may result in a prohibitive solutiontime. Now for the John Molson International Case Competition, the judge schedules have to be generated during a very short time periodbetween the evening where the draw takes place to determine the groups and the next morning when the competitions start. Thereby, wefocus our interest on metaheuristic approaches that allow generating very good solutions within reasonable computing time for large in-stances which are of practical interest.

Fig. 2. Assignment model.


3. Decomposition approach

The solution approach described in this section is a decomposition procedure where the sub-problems p associated with each round pare optimized by fixing the assignments of the other rounds to their current values. Note also the analogy with the cyclic coordinate descentmethods (Luenberger, 1984) used in the Gauss–Seidel and the Jacobi procedures to optimize a function of several variables.

The procedure can be summarized as follows: First, an initial solution is generated during Phase 1. Then, in Phase 2, a set X of sub-prob-lems are solved sequentially, and new solutions are generated by combining the solution of the sub-problem with the fixed solutions forthe other sub-problems. In other words, solving a sub-problem is equivalent to search in a part of the feasible domain where only the vari-ables of the sub-problem can be modified. The procedure terminates whenever the objective function value reaches a lower bound binf

known a priori for the problem or whenever the time elapsed reaches a specified value tempsmax. Note that binf is taken as the weightedsum of lower bounds on the number of violations of the soft constraints. We rely on the number of judges available for each round and theirgroup (lead, experienced, and new) to determine the lower bounds for the number and balance constraints. For the other soft constraints, thelower bounds are set to zero.

Algorithm 1. Decomposition procedure

Phase 1Xp, an initial solution of the sub-problem associated with round p

X :¼ XbestG :¼SP

p¼1Xp, the current and the best solution generated by the procedure

Phase 2while the stopping criterion is not met do

Identify X the set of rounds where some constraints are violatedfor each round p 2X (considered sequentially according to a random permutation) do

Fix the assignments in rounds q – p to their current values in XSolve the sub-problem associated with round p. Let X�p be the best solution obtainedX :¼ ðX � fXpgÞ [ fX�pgif X is better than XbestG then

XbestG :¼ Xend if

end forend whileXbestG is the best solution generated

The procedure described above is summarized in Algorithm 1. The current solution of the problem is denoted X ¼SP

p¼1Xp;Xp being thecurrent solution of round p. XbestG is the best solution generated so far. At each major iteration of Phase 2, we specify the set X of sub-problems including those where some constraints are violated. Note that this corresponds to an intensification strategy considering onlythe sub-problems that are more likely to improve the objective function. Once the set X is specified, the sub-problems are consideredsequentially according to a random permutation of their indices, and they are solved using an adaptation of the Tabu search method intro-duced in Lamghari and Ferland (2010).

This solution approach can also be related to the Variable Neighborhood Search (VNS) proposed by Hansen and Mladenovic (2001). In-deed, when solving a sub-problem, only the assignments for a specific round can be modified, corresponding to a specific neighborhoodstructure. However, according to the values of the assignments for the other rounds, the same specific neighborhood structure corresponds


to different subsets of the feasible domain. Furthermore, it also differs from VNS since all the sub-problems in X are solved sequentiallyeven if the best solution is improved.

4. Sub-problem formulation

Recall that 3 or 5 judges must be assigned to each individual competition j of round p and that there should be as many individual com-petitions as possible with 5 judges assigned. Thereby, we modify the structure of the problem to associate 5 destination nodes with eachindividual competition. This approach may require adding a set of dummy source nodes associated with a set If

p of fictitious judges in orderto have 5 judges assigned to each individual competition. It may also require adding a set of dummy destination nodes associated with afictitious individual competition (M + 1)p in order to assign the surplus of judges.

More specifically, let Ip = {i : dip = 1} denote the set of available judges for round p. On the one hand, if jIpjP 5M, then the number jIpj ofjudges available for round p is large enough to assign 5 judges to each individual competition and there is no need to add fictitious judges;i.e., If

p

�� ¼ 0. Furthermore, there may be judges in surplus that cannot be assigned to any individual competition if the difference (jIpj � 5M)is positive. In this case (jIpj � 5M) dummy destination nodes associated with a fictitious individual competition (M + 1)p are introduced. Onthe other hand, if jIpj < 5M, then once 3 judges are assigned to each individual competition, only jIp j�3M

2

j kindividual competitions can have

an additional pair of judges assigned. Also, there may be a judge in surplus that cannot be assigned to any individual competition if thedifference (jIpj � 3M) is an odd number; i.e., jp = (jIpj � 3M)mod2 = 1. In this case, a dummy destination node associated with the fictitiousindividual competition (M + 1)p is introduced. Furthermore, jIf

pj ¼ ð5M þ jp � jIpjÞ dummy source nodes associated with fictitious judgesare introduced in order to have 5 judges assigned to each individual competition.

In any case, we generate an assignment problem having Np source nodes (corresponding to the judges) and Np destination nodes (cor-responding to the individual competitions) where

Np ¼ jIpj þ jIfpj ¼

5M þ jp if jIpj < 5M;

jIpj otherwise

�

and the 5 destination nodes associated with individual competition j of round p are (ap + j), (ap + M + j), (ap + 2M + j), (ap + 3M + j), and(ap + 4M + j) where

ap ¼Pp�1

q¼1Nq if p P 2;

0 otherwise:

8><>:

Now let Xp denote a solution of the sub-problem associated with round p. The new mathematical model associated with this round p issummarized in Fig. 2.

The objective function (15) includes 9 different terms. The first five terms ðd1j ðXpÞ; d2

j ðXpÞ; d3j ðXpÞ; d4

irðXpÞ, and d5itðXpÞÞ are related to the

soft rules: the balance constraints (7), the affiliation constraints (8), the diversity constraints (9), the coupling constraints (10), and the se-quence constraints (11), respectively. The sixth term (dj(Xp)) is related to the admissibility constraints (3). The last three terms (gj(Xp), kj(Xp),and wj(Xp)) are related to the requirements governing the jury composition: the requirement that 3 or 5 judges must be assigned to eachindividual competition (constraints (5) and (6), and the lead and experienced judges constraints (4). The values of these terms are specifiedas follows: Denote by LjðXpÞ ¼

P4b¼0

Pi2Ip

lixiðapþbMþjÞ; EjðXpÞ ¼P4

b¼0

Pi2Ip

v ixiðapþbMþjÞ, and NjðXpÞ ¼P4

b¼0

Pi2Ipð1� li � v iÞxiðapþbMþjÞ the number

of lead judges, of experienced judges, and new judges assigned to individual competition j in solution Xp, respectively.

1. Balance constraints (7):The value of d1

j ðXpÞ should be equal to 0 if and only if, excluding the lead judge, the number of experienced and new judges assigned to jare equal. Hence,

d1j ðXpÞ ¼ ðLjðXpÞ þ EjðXpÞ � 1Þ � NjðXpÞ

�� j ¼ 1; . . . ;M:

2. Affiliation constraints (8):Let AcjðXpÞ ¼

P4b¼0

Pi2Ip

ricxiðapþbMþjÞ be the number of judges representing firm c assigned to individual competition j in solution Xp. Sincethe purpose is to have as many firms as possible in j, the d2

j ðXpÞ value is specified as follows:

d2j ðXpÞ ¼

XC

c¼1

maxðAcjðXpÞ � 1; 0Þ j ¼ 1; . . . ;M:

3. Diversity constraints (9):Denote by DkjðXpÞ ¼

P4b¼0

Pi2Ip

eikxiðapþbMþjÞ the number of judges with field of expertise k assigned to individual competition j in solution

Xp. To cover as many of the K fields of expertise as possible in j, the d3j ðXpÞ value is specified as follows:

d3j ðXpÞ ¼

XK

k¼1

maxð1� DkjðXpÞ;0Þ j ¼ 1; . . . ;M:

4. Coupling constraints (10): Let

CirðXpÞ ¼XP

q¼1

XM

j¼1

X4

b¼0

xiðaqþbMþjÞ

! X4

b¼0

xrðaqþbMþjÞ

!

¼XM

j¼1

X4

b¼0

xiðapþbMþjÞ

! X4

b¼0

xrðapþbMþjÞ

!þXq–p

XM

j¼1

X4

b¼0

xiðaqþbMþjÞ

! X4

b¼0

xrðaqþbMþjÞ

!ð19Þ


be the number of times that the pair of judges (i, r) is assigned. The coupling constraints stipulate that two judges should not be assigned

together more than once during all rounds. Consequently, the value of d4irðXpÞ should be equal to 0 if and only if Cir(Xp) 6 1. It follows that:

d4irðXpÞ ¼maxðCirðXpÞ � 1;0Þ i; r 2 Ip and r > i:

Note that since the assignments in rounds q – p are fixed, the second term on the right-hand side of Eq. (19) is a constant and needs to beevaluated only once.

5. Sequence constraints (11): Denote by

SitðXpÞ ¼XP

q¼1

XM

j¼1

X4

b¼0

stjqxiðaqþbMþjÞ ¼XM

j¼1

X4

b¼0

stjpxiðapþbMþjÞ þ constant;

the number of times that judge i is assigned to individual competitions involving team t. Then the value of d5itðXpÞ should be equal to 0 if

and only if Sit(Xp) 6 1; i.e.,

d5itðXpÞ ¼ maxðSitðXpÞ � 1;0Þ i 2 Ip and t ¼ 1; . . . ;2M:

6. Judge admissibility constraints (3):

djðXpÞ ¼X4

b¼0

Xi2Ip

ð1� aijpÞxiðapþbMþjÞ j ¼ 1; . . . ;M:

7. 3 or 5 judges assigned to each individual competition j (constraints (5) and (6): Denote by FjðXpÞ ¼P4

b¼0

Pi2If

p

PxiðapþbMþjÞ the number of

fictitious judges assigned to individual competition j in solution Xp. Then the value of gj(Xp) should be equal to 0 if and only if Fj(Xp) = 0 or2. The following expression allows satisfying this requirement:

gjðXpÞ ¼maxFjðXpÞ

2

� �� 1; FjðXpÞ � 2

FjðXpÞ2

� �� j ¼ 1; . . . ;M:

8. At least one lead judge assigned to each individual competition j (constraints (4)): The value of kj(Xp) should be equal to 0 if and only ifLj(Xp) P 1; i.e.,

kjðXpÞ ¼maxð1� LjðXpÞ; 0Þ j ¼ 1; . . . ;M:

9. At least another experienced judge, different from the lead judge, assigned to each individual competition j (constraints (4): The value ofwj(Xp) should be equal to 0 if and only if either Ej(Xp) P 1 or Lj(Xp) P 2 (since all the lead judges are experienced). The following expres-sion allows satisfying this requirement:

wjðXpÞ ¼ maxð1�max EjðXpÞ; LjðXpÞ � 1 �

;0Þ j ¼ 1; . . . ;M:

To specify properly the weights w1, . . . ,w6 used in the objective function (15), we have to note that the violations of the different con-straints are used to specify a goal programming formulation of the sub-problem. Furthermore, we account for the ordering of priority of theconstraints specified by the organizing committee of the John Molson International Case Competition (in decreasing order of priority): theadmissibility constraints and the rules related to the jury composition (w6), the sequence constraints (w5), the coupling constraints (w4), thediversity constraints (w3), the affiliation constraints (w2), and the balance constraints (w1). One approach to generate a solution for a goalprogramming problem is the sequential approach where the objective functions are considered sequentially in decreasing order of theirpriority. Each objective function is optimized providing that the values of the objective functions of higher priority do not deteriorate. In-stead of using this approach where the number of problems to solve is equal to the number of objective functions, we solve an equivalentproblem where the objective is specified according to equivalent weights for the different objective functions leading to obtain an optimalsolution for the sequential approach. Such equivalent weights based on the maximum value of each objective have been proposed by Sher-ali (1982). In this paper, we use the following equivalent weights based on both the minimum and the maximum values of each objective:

w1 ¼ 1;w2 ¼ 5M þ 1;w3 ¼ ð5M þ 1Þð4M þ 1Þ;w4 ¼ ð5M þ 1Þð4M þ 1ÞððK � 1ÞM þ 1Þ;w5 ¼ ð5M þ 1Þð4M þ 1ÞððK � 1ÞM þ 1Þð10M þ 1Þ;w6 ¼ ð5M þ 1Þð4M þ 1ÞððK � 1ÞM þ 1Þð10M þ 1Þ2:

Furthermore, it can be shown that these weights have smaller amplitude than those specified with the procedure in Sherali (1982).

5. Phase 1 to initialize the solution procedure

To determine an initial solution, the rounds p = 1, . . . ,P are considered sequentially. For each round p, we first assign randomly the Np

source nodes associated with the judges to the Np destination nodes associated with the individual competitions. Then, this initial solutionis improved by a Tabu search method attempting to reduce violations of the different assignment rules (i.e., to optimize the objective func-

tion (15)). Note that at this phase, the values of the terms d4irðXpÞ and d5

itðXpÞ related to the rules interconnecting the rounds are evaluatedaccounting only for the rounds q = 1, . . . ,p that have been considered so far.


5.1. Tabu search procedure for the sub-problem associated with a round p

To improve a current solution Xp of ðM2pÞ, we use the following Tabu search procedure. The neighborhood of a feasible solution Xp is ob-

tained by exchanging the assignment of two judges i and r currently assigned to different individual competitions j and l of round p, respec-tively. The new solution generated is denoted Xp � (i, j,r, l). Note that if the two judges i and r are fictitious (i.e., i 2 If

p and r 2 Ifp), then their

exchange has no influence on the value of the current solution. Hence, we only consider interesting exchanges (i, j,r, l) such that i R Ifp or r R If

p.To avoid cycling, attributes of recently visited solutions are declared Tabu for a random number of iterations generated in the interval

[tmin, tmax] at each iteration. Whenever we move from Xp to Xp � (i, j,r, l), both pairs (i, j) and (r, l) are introduced in the Tabu list TL. An ex-change (i, j,r, l) is considered as Tabu if (i, l) 2 TL and (r, j) 2 TL. The Tabu status of an exchange is revoked through the classical aspirationcriterion if this exchange leads to a better solution than the best solution found so far.

The first non Tabu neighbor solution yielding an improvement of the value of the current solution or the first Tabu neighbor solutionsatisfying the aspiration criterion is kept as the new current solution for the next iteration. If no such solution exists, then the best non Tabusolution in the neighborhood is selected. Note that in this case there may exist several exchanges with the same best modification value.When this occurs, we use a secondary selection criterion based on a frequency memory to break ties among the best candidate exchanges.To every candidate exchange (i, j,r, l), we assign the frequency penalty

Pði; j; r; lÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiFreqil

pþ

ffiffiffiffiffiffiffiffiffiffiffiffiFreqrj

q;

where Freqil(Freqrj) is the number of times that the pair (i, l) ((r, j)) has been involved in the selected exchanges since the beginning of thesearch. Now among the best candidate solutions Xp � (i, j,r, l), we select the one with the smallest frequency penalty value. This secondaryselection criterion can be seen as a diversification strategy used to drive the search towards less explored regions of the search space.

Exchanges are carried out until a maximal number nitermax of successive non-improving iterations is reached or until no improvementis possible; i.e., whenever all the assignment rules are met in each individual competition of round p.

6. Phase 2 to improve the solution

As mentioned in Section 3, at each iteration of Phase 2, the rounds p 2X are considered sequentially according to a random permuta-tion. Whenever a sub-problem p is considered, an initial solution X0

p is required. This solution is generated using the diversification strategydescribed in Section 6.1. Once X0

p is available, then the Tabu search method described in Section 5.1 is applied.

6.1. Diversification strategy

The purpose of a diversification strategy is to search more extensively the feasible domain. For this reason, the new initial solution isobtained by replacing some judges in some individual competitions by others that have been less frequently assigned to these competitionssince the beginning of the search. Furthermore, to explore interesting parts of the search space, these judges are chosen among those vio-lating constraints.

Denote by I(j,Xp) the set of the 5 judges assigned to individual competition j in solution Xp. We first partition the set of judges (includingthe fictitious judges) into 2 sets W and G. W is built by sequentially considering the individual competitions j:

1. If d1j ðXpÞ > 0, then include in W one of the judges i 2 I(j,Xp), being experienced or new according to the group having a surplus.

2. If d2j ðXpÞ > 0, then, for each pair of judges representing the same firm, include one of the judges i 2 I(j,Xp) in W .

3. If d3j ðXpÞ > 0, then include in W one of the judges i 2 I(j,Xp) having a field of expertise shared by others.

4. If d4irðXpÞ > 0, then include in W one of the judges i or r provided that there exists some individual competition j of round p where i and r

are assigned.5. If d5

itðXpÞ > 0, then include i in W provided that there exists some individual competition j of round p such that i 2 I(j,Xp), and the team tcompetes in j.

6. If dj(Xp) > 0, then include in W all the judges i 2 I(j,Xp) that are not admissible for j.7. If gj(Xp) > 0, then we consider the following cases according to the number Fj(Xp) of fictitious judges assigned to j:

(a) if Fj(Xp) = 1, then include in W the fictitious judge i 2 I(j,Xp)(b) if Fj(Xp) = 3, then include in W one of the fictitious judges i 2 I(j,Xp)(c) if Fj(Xp) = 4, then include in W two of the fictitious judges i, i0 2 I(j,Xp).

8. If kj(Xp) > 0, then include in W one of the judges i 2 I(j,Xp).9. If wj(Xp) > 0, then include in W one of the judges i 2 I(j,Xp).

All the other judges belong to the set G.Note that in cases (1–4, 8, and 9) where a judge is selected among a set of candidates, the selection is biased in favor of the judge less

frequently included in W up to now. For this purpose, with each judge i, we associate the following information n(i) corresponding to thenumber of times that judge i has been chosen to be included in a set W. Let qi ¼ 1

nðiÞþ1. The selection probability associated with the choice ofi among a set of judges J is then defined as

pðiÞ ¼ qiPr2Jqr

:

Once the sets W and G are identified, we determine the associated list of exchanges L which includes interesting exchanges (i, j,r, l) suchthat i and r belong to W and G, respectively. Then we use the following sequential process to generate the new initial solution. Startwith X0

p :¼ Xp. At each iteration of the process, select the exchange (i, j,r, l) in L having the smallest frequency penalty value P(i, j,r, l). Ties


are broken in favor of the exchange inducing the best modification of the objective function. Set X0p :¼ X0

p � ði; j; r; lÞ, and eliminate from L allthe exchanges involving judges i or r. The Tabu list and the frequency memory are also updated accordingly. The process terminates whenthe list L is empty.

7. Global approach

The global approach deals with the P rounds simultaneously. The initial solution is generated as in Phase 1 of the decomposition ap-proach. It is improved using a Tabu search procedure similar to the one introduced in Section 5.1 but where the assignments of any roundcan be modified rather than fixing the assignments of (P � 1) rounds and modifying only the assignments of a specific round p. Hence, theneighborhood is generated by exchanging the assignment of two judges currently assigned to different individual competitions of the sameround. When the Tabu search terminates, a diversification strategy based on similar principles to those described in Section 6 is used togenerate a new initial solution to reinitialize the solution procedure. Individual competitions of all rounds are considered when identifyingthe sets W and G.

8. Numerical results

60 different randomly generated problems are used to analyze the solution procedures introduced in this paper. They have a similarstructure as the real-world problem faced by the organizing committee of the John Molson International Case Competition, but they sim-ulate tighter situations as far as the number of available judges and their admissibility are concerned. Furthermore, to evaluate the effec-tiveness of the proposed approaches for solving large-scale problems, we also generate problems with much more individual competitionsthan those arising in the specific application of the John Molson Competition.

We consider two different sets of problems P1 and P2 where each set includes 3 different subsets (each subset including 10 differentproblems) with 15, 30, and 90 individual competitions per round, respectively. The problems are generated such that for problems inP2, there always exists a solution where all competitions have 5 judges assigned (i.e., the last term of the objective function (1) is equalto 0), and for problems in P1, some competitions have only 3 judges assigned. Note that the evaluation of the lower bound binf relies onthe structure of the problems P1 and P2 (see Lamghari (2008) for details).

The 10 random problems of each subset have identical sizes; i.e., the same number of teams and the same number of available judges.These numbers are shown in Table 1 where T and M indicate the number of teams and the number of individual competitions per round,respectively. The number of rounds P is set to the same value as in the real case (5), and consequently, the number of teams T is a multipleof 6. For each round, the number of lead judges available is equal to M and the number of experienced judges available is equal to 2M. Thenumber of new judges available is equal to M and 2M for the problems in P1 and P2, respectively. Furthermore, to make more difficult thesatisfaction of the rules interconnecting the rounds (sequence and coupling constraints), the problems are generated such that the samejudges are available for all rounds. We assume that each judge is in conflict with exactly 4 teams chosen randomly; i.e., he is not admissiblefor the individual competitions involving these teams. We also assume that 30% of the judges of each type (lead, experienced, and new) arebilingual and that 10% of the teams make their presentation in French. Like in the real application, 6 different fields of expertise for thejudges are considered. To determine the expertises of a judge, we first choose a random number a from a uniform distribution on the inter-val [1,6]. Then, a different randomly chosen numbers are the expertises for this judge. Finally, for each judge, its firm corresponds to arandom number in the interval [0,50]. The index 0 means that the judge is not affiliated to any firm (i.e., he is a university professor).The data files of the different problems are available from the authors upon request.

Note that in the John Molson context, each judge can be in conflict with at most 4 teams. Thus, the rule used to generate our problemsmakes sense in real-world terms, but these problems should be more difficult to solve. However, accordingly, we expect that the soft con-straints will be more easily satisfied when the problem size is large because a judge can be assigned to a larger number of individual com-petitions involving a much wider range of teams. Furthermore, a judge can be paired with a larger number of other judges.

Table 1Problems data.

M T Lead Experienced New

P1 P2 P1 P2 P1 P2

15 30 15 15 30 30 15 3030 60 30 30 60 60 30 6090 180 90 90 180 180 90 180

Table 2Parameters used in the solution procedures.

Name Explanation Value

TS-D TS-G

[tmin, tmax] Interval in which the duration of the Tabu status of the moves is chosen (Tabu tenure) [b0.8jIpjc,d1.2jIpje] 0:8P

pjIpjj k

; 1:2P

p jIpjl mh i

nitermax Maximum number of successive non-improving iterations before a new initial solution is generated jIpjP

pjIpj

tempsmax Maximum allotted time for the solution procedure3P

pjIp j

P seconds 3P

pjIp j

P seconds

Table 3Comparing the efficiency according to the type of the problems.

Method Set Size Ave bal Ave aff Ave div Ave coup Ave seq Ave dev %Feas %Opt Ave CPU

TS-D P1 15 4.15 3.19 5.92 2.30 2.41 4.94E10 100 0 180.3530 0 0 0 0 0 0 100 100 4.7490 0 0 0 0 0 0 100 100 11.72

P2 15 40.80 7.65 1.61 15.71 14.90 3.06E11 100 0 225.5930 0 0 0 0 0 0 100 100 16.7290 0 0 0 0 0 0 100 100 22.26

TS-G P1 15 0.54 0.71 3.61 1.96 2.57 5.27E10 100 0 183.3930 0.04 0 0.01 0 0 3.61E03 100 98 59.0690 0.02 0 0 0 0 1.24E16 79 79 969.38

P2 15 38.12 5.06 0.48 11.43 14.90 3.05E11 100 0 232.0130 1.10 0 0 0 0 2.20 100 60 372.6690 4.12 0 0 0 0 1.42E16 79 11 2 073.91

Table 4Comparing the efficiency according to the size of the problems.

Method Size Ave bal Ave aff Ave div Ave coup Ave seq Ave dev %Feas %Opt Ave CPU

TS-D 15 22.47 5.42 3.76 9.00 8.65 1.78E11 100 0 202.9730 0 0 0 0 0 0 100 100 10.7390 0 0 0 0 0 0 100 100 16.99

TS-G 15 19.33 2.89 2.05 6.70 8.74 1.79E11 100 0 207.7030 0.57 0 0.005 0 0 1.81E03 100 79 215.8690 2.07 0 0 0 0 1.33E16 79 45 1 521.64

0 50 100 150 1801011

1012

Time (sec)

Ave

dev

Problems of size 15

0 100 200 300 360

1010

Time (sec)

Ave

dev

Problems of size 30

0 100 200 300 400 500 600 700 800 900 1 000 1 080

1010

1020

Time (sec)

Ave

dev

Problems of size 90

TS−DTS−G

TS−DTS−G

TS−DTS−G

Fig. 3. Ave dev evolution during the resolution.



In the following, TS-D and TS-G denote the decomposition method and the global method, respectively. The methods are comparednumerically using the same setting of parameters as in Lamghari and Ferland (2007, 2010). The values of these parameters are indicatedin the last column of Table 2 where jIpj denotes the number of judges available for round p.

All problems are solved on a 1993 MHz processor AMD Opteron 246 with 2 GB of memory running under Linux. Each problem is solved10 times by each method using different initial solutions. Tables 3 and 4 report numerical results comparing the efficiency of the methodsaccording to the problem types and the size of the problems, respectively. The following notation is used for the comparison criteria: Avebal, Ave aff, Ave div, Ave coup, and Ave seq correspond respectively to the average deviation of the violations of the balance, affiliation, diver-sity, coupling, and sequence constraints from their corresponding lower bounds. Recall that the lower bounds are set to zero except for thebalance constraints where it is determined according to the number of judges available and their group. The average deviations related tothe number constraints are not reported since all the solutions generated by the two methods include the largest number of competitionswith 5 judges. Ave dev denotes the average deviation of the values of the solutions generated from the lower bound binf. %Feas indicates theaverage percentage of feasible solutions obtained. %Opt represents the average percentage of runs achieving binf. The average CPU time overthe 10 runs is denoted by Ave CPU and is given in seconds.

In general, the results indicate the superiority of TS-D over TS-G. The performance differences increase with the problem size. Note alsothat, according to the comments made before, the soft constraints are easier to satisfy for the larger problems.

For problems having 15 individual competitions per round, the results in Tables 3 and 4 indicate that the average performance of thetwo methods is quite similar. However, we will indicate later how their behavior differs. For problems having 30 individual competitionsper round, the results indicate that TS-D gives better results than TS-G in terms of %Opt and Ave CPU. In particular, for the problems in P2

(for which there always exists a solution where all competitions have 5 judges assigned), the %Opt is improved by 40% and the Ave CPU isreduced by a factor of 23 when the decomposition method TS-D is used. In terms of the solution quality, the results reported in Tables 3 and4 indicate that TS-G performs almost as well as TS-D since only the lower priority constraints are violated inducing a small average vio-lation. For the largest problems having 90 individual competitions per round, the global method TS-G shows a poor performance. Indeed,this method has much more difficulty to find feasible solutions. Furthermore, the average solution time increases by a factor of 90 withrespect to that of the TS-D.

Fig. 3 illustrates the behavior of the Ave dev during the resolution of the different problems. Each curve associated with a method indi-cates the average values of the Ave dev over the 10 resolutions calculated at different times of the resolution. These curves indicate evenmore clearly the superiority of TS-D over TS-G.

Even if the average performance of the two methods is quite similar, the curves indicate that TS-G requires significantly more compu-tational time to reach solutions as good as those obtained by TS-D. Furthermore, this difference in the behavior of the methods increaseswith the problem size. On the one hand, the curves indicate that after improving the solution at the beginning of the process, TS-G reaches apoint where it cannot move out of local minima. On the other hand, TS-D is able to improve rapidly the initial solution at the beginning ofthe process until reaching optimal solutions. The success of TS-D seems largely due to the fact that it focuses the search into the interestingparts of the feasible domain.

In order to complete our numerical evaluation of the two methods, we try solving a linear version of the model presented in Fig. 1 usingCPLEX 9.13. CPLEX was unable to find an integer feasible solution for any problem within 10 hours. Alternatively, we solve the linear relax-ation of the problem to obtain a lower bound on the optimal value. Again, CPLEX could not solve the largest problems with 90 individualcompetitions per round within the 10 hours time limit. It is interesting to note that the value of the lower bound obtained by CPLEX islarger (or equal) to the value binf that we use to evaluate the average deviation Ave dev. Therefore, this value of Ave dev is larger thanthe real average deviation.

Table 5Results obtained with CPLEX and with the metaheuristic methods.

Criterion Problem set Size CPLEX TS-D TS-G

%Ave Gap P1 15 NA 7.01E�03 8.07E�0330 NA 0 1.97E�1090 NA – –

P2 15 NA 0.18 0.1330 NA NA NA90 NA – –

%Feas P1 15 NA 100 10030 NA 100 10090 NA 100 79

P2 15 NA 100 10030 NA 100 10090 NA 100 79

%Opt P1 15 NA 0 030 NA 100 9890 NA 100 79

P2 15 NA 0 030 NA 100 6090 NA 100 11

Ave CPU P1 15 982.46 180.35 183.3930 893.05 4.74 59.0690 – 11.72 969.38

P2 15 350.83 225.59 232.0130 2 400.34 16.72 372.6690 – 22.26 2 073.91


In Table 5, we evaluate the effectiveness of the two methods with respect to the optimal value of the linear relaxation. For each subset ofproblems, we indicate, whenever it is possible, the value of the average relative gap %Ave Gap between the average value Ave VObj of thesolutions generated and the optimal value zLR of the linear relaxation:

AveVObj� zLR

zLR� 100:

The average percentage of feasible solutions obtained (%Feas), the average percentage of runs achieving the lower bound binf (%Opt), and theaverage CPU time (Ave CPU) in seconds are also given. The three criteria %Ave Gap, %Feas, and %Opt do not apply to CPLEX as indicated by NA.

Considering only the subsets of problems where CPLEX is able to solve the linear relaxation (subsets with 15 and 30 individual compe-titions per round), the results in Table 5 indicate that the two methods outperform CPLEX significantly. The %Ave Gap rarely exceeds 0.2%.This means that the solutions generated are very close to the optimal. With respect to the solution time, both methods require an Ave CPUsmaller than that required by CPLEX to solve the linear relaxation of the problem. In particular, for problems with 30 individual compe-titions per round, TS-D can improve the Ave CPU over CPLEX by a factor of 153.

Now, if we consider the largest problems having 90 individual competitions per round, TS-D generates always optimal solutions in lessthan 23 seconds on average, while CPLEX is not able to solve the linear relaxation of any problem within the 10 hours time limit. TS-G failsto find feasible solutions in 21% of all runs. For the other runs, TS-G is comparable to TS-D in terms of solution quality but not in terms ofsolution time.

In summary, for the tested problems, it seems to be worth using the metaheuristic approach which relies on decomposition (TS-D) sinceit can quickly generate solutions of excellent quality. Moreover, comparison showed that TS-D has the same behavior for all problem sizesand thus is more effective and robust than the alternative approach with no decomposition (TS-G).

9. Conclusion

In this paper, we study the judge assignment problem in the context of the John Molson International Case Competition. This problem isof practical interest since it is an annual recurring real-world assignment problem. We propose a decomposition approach to deal with thefive rounds problem. At each iteration, we use an adaptation of the Tabu search method introduced in Lamghari and Ferland (2010) to solvesequentially each of the one round sub-problems by fixing the assignments of the other rounds to their current values. We compare numer-ically this approach with a global approach based on a straightforward extension of the one round problem to deal simultaneously with thefive rounds. The numerical results indicate that the decomposition approach outperforms the global approach.

The decomposition approach produces very good results for the real problems arising in the specific application of the John MolsonCompetition. Currently, the organizing committee is using a software including the decomposition approach to obtain, in very short com-putational times, very high quality solutions compared to those obtained manually. Moreover, the flexibility of the approach should allowadapting it to deal with other multi-period assignment problems.

In terms of future research directions, we are interested in extending the proposed approach to solve similar problems arising in othercontexts and in developing algorithms for adapting the generated solutions to sudden and unexpected changes in data such as withdrawalof assigned judges.

Acknowledgments

The authors acknowledge the financial support of the Canadian Natural Sciences and Engineering Research Council (NSERC GrantOGP0008312). Thanks are also due to the referees for their valuable comments.

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