meta’08, october 2008referee assignment in sports leagues: bi-objective approaches 1/35 celso c....

META’08, October 2008 Referee assignment in sports leagues: bi-objective approaches

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Celso C. Ribeiro* Alexandre R. Duarte

META’08

Hammamet, October 2008

Referee assignment in sports leagues: Exact

and approximate bi-objective approaches


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Summary• Motivation• Referee assignment• Problem statement (single

criterion)– Solution approach– Numerical results

• Bi-criteria problem– Solution approach– Numerical results

• Conclusions


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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


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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


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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


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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


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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


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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 = {s1, s2,..., sn}: refereeing slots to be filled by referees

• R = {r1, r2,..., rm}: referees


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Problem statement (RAP)

• pi: skill level of referee ri • qj: minimum skill level a referee must

have to be assigned to refereeing slot sj

• Mi: maximum number of games referee ri can officiate

• Ti: target number of games referee ri is willing to officiate

• Each referee may choose a set of time slots where he/she is not available to officiate


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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


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Solution approach

• Three-phase heuristic approach to real-size instances1. Greedy constructive heuristic2. ILS-based repair heuristic to make the

initial solution feasible (if necessary)3. ILS-based procedure to improve a

feasible solution


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Solution approach

Algorithm RefereeAssignmentHeuristic (MaxIter1, MaxIter2)1. S* BuildGreedyRandomizedSolution ();2. If not isFeasible (S*) then3. S* RepairHeuristic (S*, MaxIter1);4. If isFeasible (S*) then5. S* ImprovementHeuristic (S*, MaxIter2);6. Else “infeasible”7.Return S*


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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


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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


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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


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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


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ILS-based schemeAlgorithm ILS_Scheme (S, MaxIter)1. For each facility f do2. S LocalSearch (f, S);3. For i = 1..MaxIter do4. Scurrent, f1, f2 Perturbation (S);

5. Scurrent LocalSearch (f1, Scurrent);

6. Scurrent LocalSearch (f2, Scurrent);7. S AcceptanceCriterion (S, Scurrent);8. S* UpdateGlobalBestSolution(S, S*);9. Return S*


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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)


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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


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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


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Numerical resultsInstanc

e

Construction Repair Improvement

time

(s)value feasi

bletime (s) value feasib

letime (s) value

I1 0.02 1286.20 10 — — — 32.34 619.60

I2 0.02 1360.00 5 0.47 1338.00 10 31.81 623.40

I3 0.02 1269.00 2 0.60 1247.00 10 33.87 621.60

I4 0.03 — — 1.14 1303.20 10 30.28 627.20

I5 0.03 1302.00 3 1.40 12591.14

10 33.73 654.00

Table 1: Instances with 500 games, 750 referees, 65 facilities, and pattern P0


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Numerical resultsInstan

cepattern

Greedy Random

const.

(s)repair

(s)feasible

repair

(s)feasible

I1 P00.03 11.27 10 79.80 9

I2 P00.03 6.69 10 80.80 10

I3 P00.03 11.33 10 86.20 8

I4 P00.03 4.61 10 30.60 10

I5 P00.03 3.39 10 29.10 10

I1 P10.03 2.75 10 33.50 10

I2 P10.02 19.29 10 134.60 2

I3 P10.03 14.77 10 135.10 8

I4 P10.03 1.22 10 38.00 10

I5 P10.03 2.69 10 32.90 10

Table 4: Greedy vs. randomly generated initial solutions


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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


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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”


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• Goal: find the set of (potentially) efficient solutions

• Feasible solution S’ dominates feasible solution S if z1(S’) < z1(S) and z2(S’) ≤ z2(S) or z1(S’) ≤ z1(S) and z2(S’) < z2(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)

z1

z2

Supported non-extreme efficient

Non-supported

Supported extreme efficient


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Solution approach• Exact approach: dichotomic method

– Solve lexicographically for each objective to find the two extreme efficient solutions (Sa and Sb) in the frontier

– Recursively solve a weighted sum problem, for each specific search direction (weight combination) and with two additional constraints z1(x) < z1(Sb) and z2(x) < z2(Sa), to find a solution between the two extreme points Sa and Sb

– Next direction: (z2(Sa) - z2(Sb)).z1(x) + (z1(Sb) - z1(Sa)).z2(x)

– Stop recursion when the new problem is infeasible


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Solution approach

• Exact approach: dichotomic method

50 games and 100 referees


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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)


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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)


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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


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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


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Numerical results


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Numerical results


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Numerical results


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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


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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


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Repair heuristic

• Perturbation– Select a facility fk 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 ri that officiates at another facility (or does not officiate at all) whose skill level is at least as large as the above


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Repair heuristic

• Perturbation (cont.)– Randomly select referees that officiate

at the same facility to be assigned to the refereeing slots currently assigned to ri

– Finally, assign referee ri to a refereeing slot at facility fk which is currently assigned to a referee with at least one violation


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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


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Enhancements1. 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)


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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

• MaxIter1 = 10,000 (repair heuristic) MaxIter2 = 200 (improvement heuristic)


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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


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Instance

patternCPLEX Heuristic

optimum time (s) average

best time(s)

I2 P043 164.10 47.0 44 18.99

I3 P018 200.20 20.8 18 3.05

I5 P044 137.70 45.2 44 11.45

I2 P165 128.20 67.2 65 15.32

I5 P172 47.40 82.4 75 8.83

Table 5: 33 games, 57 referees, and five facilities


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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


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Time-to-target (TTT) plots

500 games, 750 referees, 85 facilities , and pattern P0 (target = 529)

200 runs for given target


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500 games, 750 referees, 65 facilities, and pattern P1 (target = 926)


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Solution approach (biRAP)

• Exact approach: dichotomic method

50 games and 100 referees

(largest instance exactly solved)


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• 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 1.000 ILS perturbations• every 200 iterations, before 2.000 ILS perturbations,

...

– Path relinking used as a post-optimization procedure


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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


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• 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

meta’08, october 2008referee assignment in sports leagues: bi-objective approaches 1/35 celso c....

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