Models and Methods
for
the Judge Assignment Problem
Amina Lamghari
Jacques A. Ferland
Computer Science & OR Dept.
University of Montreal
Problem Background
• The John Molson MBA International Case Competition
• Takes place every year for the last 35 years at Concordia University
• 30 teams of business school students coming from top international
universities
• Partitioned in 5 groups of 6 teams
• First part of the Competition is a round-robin tournament (5 rounds)
• At each round, each team competes against each of the other 5 teams of its
group
• The three best teams move to the finals in the second part of the
Competition
• Problem: Judge Assignment to the competitions of the first part
Judge Assignment to the Competitions of a Round
Hard rules that must be satisfied
•
A judge cannot be assigned to a competition involving a team
coming from a University where he received his degree
•
A judge cannot be assigned to a competition involving a team
coming from a University where he is a faculty member
•
3 or 5 judges must be assigned to each competition
•
At least one of these judges belongs to the set of lead judges
Constraints
Soft rules to be satisfied as much as possible
•
•
The expertises of the judges assigned to a competition should be as
different as possible to cover as many of the 6 fields of expertise as
possible
The number of competitions having 5 judges assigned should be
maximized
Objective function
Mathematical Formulation
Number of judges with
expertise k assigned to j
Minimize the number of
competitions with 3 judges
A judge cannot be assigned to a competition involving a team
Subject to
coming from a University where he received his degree
A judge cannot be assigned to a competition involving a team
coming from a University where he is a faculty member
aij=1 iff i admissible for j
At least one lead judge
3 or 5 judges assigned
Solution Approach
• A metaheuristic approach based on Tabu Search
Tabu Search is a metaheuristic that guides a local search procedure to
explore the solution space allowing to move out of local minimums
 Allows moving to neighbor solutions that can deteriorate the current objective
function value
 Uses a short-term memory (Tabu list) to forbid returning to recently visited
solutions
• Extend the solution space to allow infeasible solutions
Judge Assignment for Several Rounds
Hard rules that must be satisfied
•
A judge cannot be assigned to a competition
involving a team coming from a University where he
received his degree
•
A judge cannot be assigned to a competition
involving a University where he is a faculty member
•
A judge cannot be assigned to a competition
involving a team that he does not wish to evaluate
•
If a team in a competition is presenting in
French, then the judges assigned to this competition
must be fluent in French
•
3 or 5 judges must be assigned to each
competition
Soft rules to be satisfied as much as
possible
•
The judges assigned to each competition
should be balanced with regard to the number of
experienced and new judges
•
If several judges coming from firms are
assigned to a competition then they should come
from different ones
•
The expertises of the judges assigned to a
competition should cover as many of the 6 expertises
as possible
•
The number of competitions having 5 judges
assigned should be maximized
•
At least one of these judges belongs to the set of
lead judges
Gl Global constraints connecting the rounds
•
A specific pair of judges should not be assigned
more than once during all the rounds
•
At least one experienced judge, different from
the lead judge, must be assigned to each
competition
•
During the different rounds, a judge should
not be assigned to different competitions involving
the same team
Minimize the weighted sum of
the violations of the soft rules
A judge cannot be assigned to a competition involving a team
coming from a University where he received his degree
A judge cannot be assigned to a competition involving a team
coming from a University where he is a faculty member
aijp =1 iff i admissible for j
A judge cannot be assigned to a competition involving a team
At least one lead judge and
that he does not wish to evaluate
at least one experienced judge
If a team in a competition is presenting in French,
then the judges 3assigned
to thisassigned
competition must
or 5 judges
be fluent in French
number of experienced = number of new judges
each firm at most once in each competition
each field at least once
in rules
each competition
Soft
each pair of judges at most once in rounds
same judge to different competitions of a team
Deviation variables: number of
violations of each soft rule
Metaheuristic Solution Approach
• Sequential procedure based on Tabu Search
• Includes two major phases
Phase1 (initialization)
 For each round p (p=1,…,5) individually, determine an initial solution xinitp accounting
for the global constraints connecting the rounds as they become relevant
 xbest := Up xinitp the best solution generated so far
 xLast := Up xinitp the best current solution
Phase 2 (iterative process repeated until the stopping criterion is met)
 Generate a permutation P of (1,…,5)
 For each round p in P
 Determine a new initial solution x0p
 Determine a new local optimal solution x*p
 xLast := (xLast – xLastp) U x*p
 If xLast is better than xbest, then xbest := xLast