META’08, October 2008Referee assignment in sports leagues: bi-objective approaches 1/35 Celso C. Ribeiro* Alexandre R. Duarte META’08 Hammamet, October 2008 Referee assignment in sports leagues: Exact and approximate bi- objective approaches

META’08, October 2008Referee assignment in sports leagues: bi-objective approaches 2/35 Summary Motivation Referee assignment Problem statement (single criterion) –Solution approach –Numerical results Bi-criteria problem –Solution approach –Numerical results Conclusions

META’08, October 2008Referee assignment in sports leagues: bi-objective approaches 3/35 Motivation Optimization in sports is a field of increasing interest –Traveling tournament problem –Playoff elimination –Tournament scheduling –Referee assignment Regional amateur leagues in the US (baseball, basketball, soccer): hundreds of games every weekend in different divisions

META’08, October 2008Referee assignment in sports leagues: bi-objective approaches 4/35 Motivation Single league in California: up to 500 soccer games in a weekend, to be refereed by hundreds of certified referees MOSA League (Monmouth & Ocean Counties Soccer Association, NJ): boys & girls, ages 8-18, six divisions per age/gender group, six teams per division: 396 games every Sunday (US$ 40 per ref; U$ 20 per linesman, two linesmen are needed) Problem: assign referees to games Many constraints and multiple objectives involved

META’08, October 2008Referee assignment in sports leagues: bi-objective approaches 5/35 Referee assignment Possible constraints: –Number of referees may vary, depending on the sport, league, and tournament –Games may require referees with different levels of certification: games in higher divisions require referees with higher skills –A referee cannot be assigned to a game where he/she is a player –Timetabling conflicts and traveling times

META’08, October 2008Referee assignment in sports leagues: bi-objective approaches 6/35 Referee assignment Possible constraints (cont’d): –Referee groups: cliques of referees that request to be assigned to the same games (relatives, car pools) Hard links Soft links –Number of games a referee is willing to referee –Traveling constraints –Some referees can officiate games only at a certain location or period of the day

META’08, October 2008Referee assignment in sports leagues: bi-objective approaches 7/35 Referee assignment Possible objectives: –Difference between the target number of games the referee is willing to officiate and the number of games he/she is assigned to –Traveling or idle time between consecutive games –Number of inter-facility travels –Number of games assigned outside his/her preferred time-slots or facilities –Number of violated soft/hard links

META’08, October 2008Referee assignment in sports leagues: bi-objective approaches 8/35 Problem statement (RAP) Games are already scheduled (facility – time slot) Each game has a number of refereeing positions to be assigned to referees Each refereeing position to be filled by a referee is called a refereeing slot S = {s 1, s 2,..., s n }: refereeing slots to be filled by referees R = {r 1, r 2,..., r m }: referees

META’08, October 2008Referee assignment in sports leagues: bi-objective approaches 9/35 Problem statement (RAP) p i : skill level of referee r i q j : minimum skill level a referee must have to be assigned to refereeing slot s j M i : maximum number of games referee r i can officiate T i : target number of games referee r i is willing to officiate Each referee may choose a set of time slots where he/she is not available to officiate

META’08, October 2008Referee assignment in sports leagues: bi-objective approaches 10/3 5 Problem statement (RAP) Problem: assign a referee to each refereeing slot Constraints (set partitioning formulation): –Referees officiate in a single facility on the same day –Minimum skill level requirements –Maximum number of games –Timetabling conflicts and availability Objective: minimize the sum over all referees of the absolute value of the difference between the target and the actual number of games assigned to each referee

META’08, October 2008Referee assignment in sports leagues: bi-objective approaches 11/3 5 Solution approach Three-phase heuristic approach to real-size instances 1.Greedy constructive heuristic 2.ILS-based repair heuristic to make the initial solution feasible (if necessary) 3.ILS-based procedure to improve a feasible solution

META’08, October 2008Referee assignment in sports leagues: bi-objective approaches 12/3 5 Solution approach Algorithm RefereeAssignmentHeuristic (MaxIter 1, MaxIter 2 ) 1. S*  BuildGreedyRandomizedSolution (); 2. If not isFeasible (S*) then 3. S*  RepairHeuristic (S*, MaxIter 1 ); 4. If isFeasible (S*) then 5. S*  ImprovementHeuristic (S*, MaxIter 2 ); 6. Else “infeasible” 7.Return S*

META’08, October 2008Referee assignment in sports leagues: bi-objective approaches 13/3 5 Greedy constructive heuristic Assign refereeing slots to referees that are also players While there are unassigned refereeing slots and unassigned referees do: –Select the highest skill level (hSL) among all unassigned referees –Greedily select the facility with unassigned slots with the strongest need for a referee with skill level hSL –Assign as many refereeing slots (games) in this facility as possible to a referee with skill level hSL Complete the solution with infeasible assignments

META’08, October 2008Referee assignment in sports leagues: bi-objective approaches 14/3 5 Local search and neighborhoods Solutions built by a construction algorithm are not necessarily optimal Local search algorithm successively replaces the current solution by a better one in a neighborhood of the first, terminating at a local optimum First improving strategy: current solution is replaced by the first neighbor whose cost function value improves that of the current solution

META’08, October 2008Referee assignment in sports leagues: bi-objective approaches 15/3 5 Local search and neighborhoods Swap moves: referees assigned to two refereeing slots are swapped (number of games assigned to each referee does not change) Exchange moves: referee assigned to a refereeing slot is replaced by another referee (number of games assigned to each referee either increases or decreases by one unit) Only moves involving referees that officiate at the same facility (or do not officiate at all) are allowed

META’08, October 2008Referee assignment in sports leagues: bi-objective approaches 16/3 5 ILS-based scheme Both the repair and the improvement heuristics use similar ILS (iterated local search) schemes: –Apply first improving local search (first exchange moves, next swap moves) to the initial solution –For a given number of iterations, apply a perturbation to the current solution, followed by local search considering only the facilities involved in the perturbation, and accept the new solution if it is better than the current one

META’08, October 2008Referee assignment in sports leagues: bi-objective approaches 17/3 5 ILS-based scheme Algorithm ILS_Scheme (S, MaxIter) 1. For each facility f do 2.S  LocalSearch (f, S); 3. For i = 1..MaxIter do 4.S current, f 1, f 2  Perturbation (S); 5.S current  LocalSearch (f 1, S current ); 6.S current  LocalSearch (f 2, S current ); 7.S  AcceptanceCriterion (S, S current ); 8.S*  UpdateGlobalBestSolution(S, S*); 9. Return S*

META’08, October 2008Referee assignment in sports leagues: bi-objective approaches 18/3 5 Repair heuristic Referees do not travel in the solutions built by the greedy constructive heuristic Possibly violated constraints: –timetabling conflicts –referee availability –skill levels –maximum number of games Approach: minimize the number of violations (no violations in feasible solutions)

META’08, October 2008Referee assignment in sports leagues: bi-objective approaches 19/3 5 Improvement heuristic Minimize the sum over all referees of the difference between the target and the actual number of games assigned to each of them Only exchange moves and perturbations that maintain feasibility are considered Swap moves are not applied, since they cannot reduce the value of the objective function

META’08, October 2008Referee assignment in sports leagues: bi-objective approaches 20/3 5 Numerical results Randomly generated instances following patterns similar to real-life applications Instances with up to 500 games and 1,000 referees Initial solutions: –random assignments (to test the repair heuristic) –greedy constructive heuristic

META’08, October 2008Referee assignment in sports leagues: bi-objective approaches 21/3 5 Numerical results Instance ConstructionRepairImprovement time (s)valuefeasib le time (s)valuefeasibl e time (s)value I1I ——— I2I I3I I4I40.03—— I5I Table 1: Instances with 500 games, 750 referees, 65 facilities, and pattern P 0

META’08, October 2008Referee assignment in sports leagues: bi-objective approaches 22/3 5 Numerical results Instanc e pattern GreedyRandom const. (s) repair (s)feasibl e repair (s) feasib le I1I1 P0P I2I2 P0P I3I3 P0P I4I4 P0P I5I5 P0P I1I1 P1P I2I2 P1P I3I3 P1P I4I4 P1P I5I5 P1P Table 4: Greedy vs. randomly generated initial solutions

META’08, October 2008Referee assignment in sports leagues: bi-objective approaches 23/3 5 Bi-criteria problem (biRAP) Same constraints as in the single objective version Objectives: 1.minimize the sum over all referees of the absolute value of the difference between the target and the actual number of games assigned to each referee 2.minimize the sum over all referees of the total idle time between consecutive games

META’08, October 2008Referee assignment in sports leagues: bi-objective approaches 24/3 5 Bi-criteria problem (biRAP) Formulation: bi-criteria set partitioning problem Variables: possible “routes” for each referee (stops correspond to refereeing positions) Each “route” has at most 4 to 5 stops: number of variables is limited Constraints: –each refereeing position has to be filled by exactly one qualified referee –each referee must perform exactly one “route”

META’08, October 2008Referee assignment in sports leagues: bi-objective approaches 25/3 5 Goal: find the set of (potentially) efficient solutions Feasible solution S’ dominates feasible solution S if z 1 (S’) < z 1 (S) and z 2 (S’) ≤ z 2 (S) or z 1 (S’) ≤ z 1 (S) and z 2 (S’) < z 2 (S) (minimization) Feasible solution S is an efficient solution if and only if there is no other feasible solution S’ that dominates S Bi-criteria problem (biRAP) z1z1 z2z2 Supported non-extreme efficient Non- supported Supported extreme efficient

META’08, October 2008Referee assignment in sports leagues: bi-objective approaches 26/3 5 Solution approach Exact approach: dichotomic method –Solve lexicographically for each objective to find the two extreme efficient solutions (S a and S b ) in the frontier –Recursively solve a weighted sum problem, for each specific search direction (weight combination) and with two additional constraints z 1 (x) < z 1 (S b ) and z 2 (x) < z 2 (S a ), to find a solution between the two extreme points S a and S b –Next direction: ( z 2 (S a ) - z 2 (S b ) ).z 1 (x) + ( z 1 (S b ) - z 1 (S a ) ).z 2 (x) –Stop recursion when the new problem is infeasible

META’08, October 2008Referee assignment in sports leagues: bi-objective approaches 27/3 5 Solution approach Exact approach: dichotomic method 50 games and 100 referees

META’08, October 2008Referee assignment in sports leagues: bi-objective approaches 28/3 5 Solution approach Heuristic approach: –Perform three-phase ILS-based heuristic for a fixed number of search directions –Each search direction represents a set of weights associated with each objective –Directions are chosen as in the dichotomic method –All new potentially efficient solutions found during the search are progressively stored –Former potentially efficient solutions are discarded during the search (quadtree is used)

META’08, October 2008Referee assignment in sports leagues: bi-objective approaches 29/3 5 Solution approach Path relinking: post-optimization procedure –If there are N efficient solutions in the current Pareto frontier, then apply path relinking to N pairs of efficient solutions –Randomly select two efficient solutions from the current Pareto frontier (initial and target) –Assignments of referees officiating at the same facility in both the initial and the target solutions will not be changed (local search in restricted neighborhood)

META’08, October 2008Referee assignment in sports leagues: bi-objective approaches 30/3 5 Solution approach Path relinking: post-optimization procedure –PR moves: select two referees to have all their assignments exchanged (at least one of them should move to the same facility where he/she officiates in the target solution) –Select the move associated with the pair of referees leading to the lowest value of the weighted sum –After path relinking, apply once again the ILS heuristic to the best solution visited in the path –Back and forward path relinking

META’08, October 2008Referee assignment in sports leagues: bi-objective approaches 31/3 5 Numerical results Parameters: –31 search directions are used –Local search stops for each direction after 1800 seconds Results illustrated for one instance with 50 referees and 100 games

META’08, October 2008Referee assignment in sports leagues: bi-objective approaches 32/3 5 Numerical results

META’08, October 2008Referee assignment in sports leagues: bi-objective approaches 33/3 5 Numerical results

META’08, October 2008Referee assignment in sports leagues: bi-objective approaches 34/3 5 Numerical results

META’08, October 2008Referee assignment in sports leagues: bi-objective approaches 35/3 5 Conclusions Optimization in sports is a field of increasing interest New optimization problem in sports Effective heuristics: construction, repair, improvement, path relinking Quick procedures to build good initial solutions Bicriteria approach is promising and finds good approximations of the Pareto frontier Other constraints and criteria may be considered

META’08, October 2008Referee assignment in sports leagues: bi-objective approaches 36/3 5 Greedy constructive heuristic Assign refereeing slots to referees that are also players While there are unassigned refereeing slots and unassigned referees do: –Select the highest skill level (hSL) among all unassigned referees –Greedily select the facility with unassigned slots with the strongest need for a referee with skill level hSL –Assign as many refereeing slots (games) in this facility as possible to a referee with skill level hSL Complete the solution with infeasible assignments Only feasibility issues are taken into account

META’08, October 2008Referee assignment in sports leagues: bi-objective approaches 37/3 5 Repair heuristic Perturbation –Select a facility f k with infeasible assignments –Select the maximum minimum skill level over all refereeing slots in this facility assigned to referees with at least one violation –Search for a referee r i that officiates at another facility (or does not officiate at all) whose skill level is at least as large as the above

META’08, October 2008Referee assignment in sports leagues: bi-objective approaches 38/3 5 Repair heuristic Perturbation (cont.) –Randomly select referees that officiate at the same facility to be assigned to the refereeing slots currently assigned to r i –Finally, assign referee r i to a refereeing slot at facility f k which is currently assigned to a referee with at least one violation

META’08, October 2008Referee assignment in sports leagues: bi-objective approaches 39/3 5 Improvement heuristic Perturbation –Select two referees officiating at different facilities, such that the swap of all their assignments is feasible –Look ahead: check if there are other games that could be officiated by these referees after the swap Only refereeing slots assigned to referees that are officiating too many games (more than their target) are considered (exchange moves involving other referees cannot reduce the objective function) –The first pair of referees whose swap can reduce the objective function is selected and all their assignments are swapped

META’08, October 2008Referee assignment in sports leagues: bi-objective approaches 40/3 5 Enhancements 1.Greedy constructive heuristic (emphasis on cost): –First, assign each referee to a number of refereeing slots as close as possible to his/her target number of games –Second, if there are still unassigned slots, assign more games to each referee 2.MIP-based improvement heuristic: –After each perturbation, instead of applying a local search for both facilities involved in this perturbation, solve a MIP model associated with the sub-problem considering all refereeing slots and referees corresponding to these facilities (“MIP it!”, Fischetti)

META’08, October 2008Referee assignment in sports leagues: bi-objective approaches 41/3 5 Numerical results (RAP) Randomly generated instances following patterns similar to real-life applications Instances with up to 500 games and 1,000 referees –Different number of facilities –Different patterns of the target number of games Five different instances for each configuration MaxIter 1 = 10,000 (repair heuristic) MaxIter 2 = 200 (improvement heuristic)

META’08, October 2008Referee assignment in sports leagues: bi-objective approaches 42/3 5 Numerical results (RAP) Codes implemented in C Results on a 2.0 GHz Pentium IV processor with 256 Mbytes of RAM Initial solutions: –random assignments (to test the repair heuristic) –greedy constructive heuristic (emphasis on feasibility) –greedy constructive heuristic (emphasis on cost) Improvement heuristic: –Neighborhood-based ILS –MIP-based ILS

META’08, October 2008Referee assignment in sports leagues: bi-objective approaches 43/3 5 Numerical results (RAP) Instancepattern CPLEXHeuristic optimumtime (s)averagebesttime(s) I2I2 P0P I3I3 P0P I5I5 P0P I2I2 P1P I5I5 P1P Table 5: 33 games, 57 referees, and five facilities

META’08, October 2008Referee assignment in sports leagues: bi-objective approaches 44/3 5 Time-to-target (TTT) plots Select an instance and a target value Perform 200 runs using different seeds Stop when a solution at least as good as the target value is found For each run, measure the time-to- target Plot the probabilities of finding a solution at least as good as the target value within some time TTT plots can illustrate algorithm robustness and are very useful for comparisons based on the probability distribution of the time-to-target

META’08, October 2008Referee assignment in sports leagues: bi-objective approaches 45/3 5 Time-to-target (TTT) plots 500 games, 750 referees, 85 facilities, and pattern P 0 (target = 529) 200 runs for given target

META’08, October 2008Referee assignment in sports leagues: bi-objective approaches 46/3 5 Numerical results (RAP) 500 games, 750 referees, 65 facilities, and pattern P 1 (target = 926)

META’08, October 2008Referee assignment in sports leagues: bi-objective approaches 47/3 5 Solution approach (biRAP) Exact approach: dichotomic method 50 games and 100 referees (largest instance exactly solved)

META’08, October 2008Referee assignment in sports leagues: bi-objective approaches 48/3 5 Solution approach (biRAP) Heuristic approach (cont’d): –Hybrid ILS approach: neighborhood-based ILS, performing a MIP exact resolution (very slow) for each facility after a fixed number of iterations (more frequently in the beginning, when the local search finds new efficient solutions more often) every 100 iterations, before ILS perturbations every 200 iterations, before ILS perturbations,... –Path relinking used as a post-optimization procedure

META’08, October 2008Referee assignment in sports leagues: bi-objective approaches 49/3 5 Solution approach (biRAP) Path relinking (cont’d): –Evaluate every move: select two referees to have all their assignments exchanged (at least one of them should move to the same facility where he/she officiates in the target solution) –For each exchange move preserving feasibility, temporarily perform the exchange followed by local search –Select the move associated with the pair of referees leading to the lowest value of the weighted sum

META’08, October 2008Referee assignment in sports leagues: bi-objective approaches 50/3 5 Solution approach (biRAP) Path relinking (cont’d): –After path relinking, apply once again the hybrid ILS heuristic to the best solution visited in the path –Perform path relinking twice for each pair of efficient solutions, exchanging the roles of the initial and target solutions and using the search direction (weight combination) associated with the initial solution