Assigning Judges to Competitions

UsingTabu Search Approach

Amina Lamghari Jacques A. Ferland Computer science and OR dept. University of Montreal

Faculty of Information Technology

University of Science

Vietnam National University of Ho Chi Minh City

March 2010

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 withexpertise k assigned to j

aij = 1 iff i admissible for j

1, 3 or 5 judges assigned

Penalty for competitions with 1 judge

Metaheuristic Solution Approach

Initial Solution

First StageStructured Neighborhood Tabu Search to reducethe number of competitions with 1 or 3 judges

Second StageTabu Search to improve the diversity of the fields of expertise of the judges assigned to a competition

Diversification strategyAdaptive 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

),,,( ljrix ),,,( ljrix

•

Structured NeighborhoodReassignment of pair (i, r) from comp. j to comp. l

from j to lImpact on the objective function

improving

improving or deteriorating

deteriorating

•

Search Strategy

from j to lImpact on the objective function

improving

Using sequentially V1(x),V2(x), V3(x).Exhaustive search

•

Search Strategy

improving or deteriorating

deteriorating

Using sequentially V4(x),…, V8(x).

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 StrategySimilarity 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 j to lImpact 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

),,,( ljrix

),,,( ljrix

),,,( ljrix

• 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

),,,( lrjix

),,,( lrjix

),,,( lrjix

• 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

in randomly selected xbestxx

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

in randomly selected xbestxx

otherwise.

if

),(

),(),( 0

xjI

MjxbestjIxjI best

xjii

xjIsolution in n competitio

individual toassigned is judge:),(

• 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

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

Same pair of judges cannot be assigned more than once

Same judge cannot be assigned more than once at

different competitions of any given team

Judge Asssignment for the 5 rounds

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

At least one experienced judge, different from the lead judge,

must be assigned to each competition

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

Phase 1 For each round 1, ,5 considered individually, determine

an initial solution accounting for the pairs of judges already

assigned for the other rounds already completed

p

p p

xinit

5

1

5

1

: the best solution generated so far

: the best current solution

pp

pp

xbest xinit

xlast xinit

Solution procedure including two phases:

Phase 2: Iterative process to improve the current

Phase 1: Initialisation to generate an initial so

____________

lution

______

_

so

__

lution

_____

Phase 2 (Iterative process repeated until a stopping criterion is met)At each iteration of the process:

Use the earlier procedure using

tabu searches and diversification

adapted to account for the

additional constraints

*: p pxlast xlast xlast x If is better than , then :xlast xbest xbest xlast

For each round ,p P

Generate a permutation of the the rounds

where some constraints are violated

P

Fix the assignments of the judges in all the other rounds*Determine a new optimal solution for round px p

Another procedure also using

tabu search and diversification

have also been developed.

Gauss-Southwell type approach to

solve the global 5 rounds problem

allowing to modify only the assignments

for each round sequentially

Having the global problem in view,

solving the round sub-problem is

equivalent to consider neighborhood

for the global problem where only

assignments of round can be modi . fied

p

p

Interpretation underlying an iteration:

Variable neighborhood type procedure

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.

•

CPLEX

•

Worth using metaheuristic

CPLEX much more CPUHeuristic very small Ave dev

•

Ave dev < 1

At most one competition where 2judges have the same expertise

•

Robustness

H-Best, H-First, R-First: Optimal value or lower bound achieved forat 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 solutionsH-Best: smaller CPU

•

H-First vs H-Best

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

H-First vs H-Best

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

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

•

H-First vs H-BestH-First: better solutionsH-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