Assigning Judges to Competitions Using Tabu Search Approach Amina Lamghari Jacques A. Ferland Computer science and OR dept. University of Montreal

Problem background The John Molson Business School International Case Competition Takes place every year for more than 30 years at Concordia University in Montreal 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 including 5 rounds where each team competes against each of the other 5 teams of its group The three best teams move to the finals

Judge Assignment to the competitions of a round. Constraints Objective function

Maximize the number of competitions with 5 judges At least one lead judge 3 or 5 judges assigned Number of judges with expertise k assigned to j a ij = 1 iff i admissible for j 1, 3 or 5 judges assigned Penalty for competitions with 1 judge

Metaheuristic Solution Approach Initial Solution First Stage Structured Neighborhood Tabu Search to reduce the number of competitions with 1 or 3 judges Second Stage Tabu Search to improve the diversity of the fields of expertise of the judges assigned to a competition Diversification strategy Adaptive memory of best solutions generated Crossover to generate a new initial solution

Initial Solution Two different processes i) Random Assign randomly 1 lead judge to each competition Hence all competitions have 1 judge assigned ii) HLA-HOA Constructive Heuristic 1. Assign 1 lead judge to each competition 2. Assign a first pair of additional judges to each competition 3. Assign a second pair of additional judges to as many competitions as possible Look ahead features making further assignments easier.

First Stage Structured Neighborhood Tabu Search Neighborhood Reassignment of a pair of judges (i, r) from competition j to competition l : in the neighborhood if the solution is feasible

Structured Neighborhood Reassignment of pair (i, r) from comp. j to comp. l from jto l Impact on the objective function improving improving or deteriorating deteriorating

Search Strategy from jto l Impact on the objective function improving Using sequentially V 1 (x), V 2 (x), V 3 (x). Exhaustive search

Search Strategy improving or deteriorating deteriorating Using sequentially V 4 (x), …, V 8 (x). No exhaustive search. After any reassignment in V 4 (x), return to V 2 (x). After any reassignment in V k (x), k = 5, …, 8, return to V 1 (x).

Search Strategy Similarity with Variable Neighborhood Search (VNS) But the search strategy strongly depend on the partition of the neighborhood, and the potential improvement associated with the different subsets from jto l Impact on the objective function improving improving or deteriorating deteriorating

Tabu list When becomes the current solution (i, j) and (r, j) are included in the Tabu list Tabu solution is Tabu if (i, l) and (r, l) are in the Tabu list Aspiration criterion satisfies the aspiration criterion if its value is better than the best value reached so far

Selection of the new current solution in the neighborhood Two different strategies compared numerically Best Generate the entire neighborhood and select the best solution in it First During the generation of the neighborhood, select - the first non Tabu solution improving the value of the current solution or - the first solution satisfying the aspiration criterion

Stopping criteria - No improvement possible 5 judges assigned to each competition or 3 or 5 judges assigned to each competition and only 0 or 1 judge is not assigned - nitermax successive iterations where the objective function does not improve

Second Stage Tabu Search Neighborhood Exchange judge i of competition j and judge r of competition l in the neighborhood if the solution is feasible Tabu list When becomes the current solution (i, j) and (r, l) are included in the Tabu list Tabu solution is Tabu if (i, l) and (r, j) are in the Tabu list

Aspiration criterion (idem) Selection of the new current solution in the neighborhood (idem) Stopping criteria - No improvement possible in each competition, all judges have different expertise or lower bound known a priori for the problem is reached - nitermax successive iterations where the objective function does not improve

Diversification Strategy Adaptive Memory Γ: set of best solutions found so far Uniform crossover Uniform crossover of and xbest to generate x 0 % of elements from xbest decreases with the number of recent successive major iterations where the objective does not improve

Diversification Strategy Adaptive Memory Γ: set of best solutions found so far Uniform crossover Uniform crossover of and xbest to generate x 0 % of elements from xbest decreases with the number of recent successive major iterations where the objective does not improve

Repair process Eliminate duplicate judges from x 0 Bias in favour of elements from xbest Look ahead feature to have a lead judge in each competition Assign a lead judge to those competitions missing one For competition having 2 or 4 judges Assign an admissible currently non assigned judge if possible Otherwise eliminate 1 judge (making sure that the competition has a lead judge assigned) New initial solution x 0 is used as a new initial solution for the next major iteration

Numerical Results 4 variants: H-Best, R-Best, H-First, R-First Initial solution : Heuristic (H), Random (R) Selection strategy: Best, First 3 sets of randomly generated problems P 1, P 2, P 3 In each set: subsets (10 problems) with 15, 50, 150, and 500 competitions P 1 : some comp. with 3 judges; judges with diff. expertise in all comp. P 2 : all comp. with 5 judges; in some comp.,judges with same expertise P 3 : some comp. with 3 judges; in some comp.,judges with same expertise

Worth using metaheuristic CPLEX: much more CPU Variants: very small Ave dev Robustness Variants: solutions of excellent quality for all problems Ave dev < 1 At most one competition where 2 judges have the same expertise

Constraints Objective function

Numerical Results 2 sets of randomly generated problems P 1, P 2 P 1 : some competitions with 3 judges P 2 : all competitions with 5 judges In each set: subsets (10 problems) with 15, 30, and 90 competitions per round Worth using metaheuristic CPLEX: fails to find an integer feasible solution in 10 hours Variants: solutions of good quality in less than 10 seconds

Software Solution approach embedded into a user friendly software Results on real data to the full satisfaction of organizing committee.

CPLEX

Worth using metaheuristic CPLEX much more CPU Heuristic very small Ave dev

Ave dev < 1 At most one competition where 2 judges have the same expertise

Robustness H-Best, H-First, R-First: Optimal value or lower bound achieved for at least one solution out of 5 R-Best: Also verified except for 2 instances of P 3 with 150 comp., and 1 of P 3 with 500 comp.

Heuristic initial solution is better H-Best dominates R-Best H-First dominates R-First

H-First vs H-Best H-First: better solutions

H-First vs H-Best H-First: better solutions H-Best: smaller CPU

H-First vs H-Best Problem of size 150 H-First: better solutions factor 7 H-Best: smaller CPU factor 2

H-First vs H-Best Problem of size 150 H-First: better solutions factor 7 H-Best: smaller CPU factor 2 Problem of size 500 H-First: better solutions factor 2 H-Best: smaller CPU factor 4

H-First vs H-Best H-First: better solutions H-Best: smaller CPU factor increasing with problem size

R-First vs R-Best Similar results

Conclusion All variants generate solutions of excellent quality With regards to CPU: H-Best is slightly dominating