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Assigning Judges to Competitions Using Tabu Search Approach Amina Lamghari Jacques A. Ferland Computer science and OR dept. University of Montreal. Problem background.
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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
Penalty for competitions with 1 judge Maximize the number of competitions with 5 judges Number of judges with expertise k assigned to j aij = 1 iff i admissible for j At least one lead judge 3 or 5 judges assigned 1, 3 or 5 judges assigned
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 StageStructured 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 improving improving or deteriorating deteriorating from j to l Impact on the objective function
Search Strategy improving Using sequentially V1(x), V2(x), V3(x). Exhaustive search from j to l Impact on the objective function
Search Strategy Using sequentially V4(x), …, V8(x). improving or deteriorating deteriorating No exhaustive search. After any reassignment in V4(x), return to V2(x). After any reassignment in Vk(x), k = 5, …, 8, return to V1(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 improving improving or deteriorating deteriorating from j to l Impact on the objective function
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 x0 % 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 x0 % 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 x0 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 x0 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 P1, P2, P3 In each set: subsets (10 problems) with 15, 50, 150, and 500 competitions P1: some comp. with 3 judges; judges with diff. expertise in all comp. P2: all comp. with 5 judges; in some comp.,judges with same expertise P3: some comp. with 3 judges; in some comp.,judges with same expertise
Worth using metaheuristic CPLEX: much more CPU Variants: very small Ave dev Ave dev < 1 At most one competition where 2 judges have the same expertise Robustness Variants: solutions of excellent quality for all problems
Constraints Objective function
Numerical Results 2 sets of randomly generated problems P1, P2 P1: some competitions with 3 judges P2: 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.
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 P3 with 150 comp., and 1 of P3with 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