MgtOp 470—Business Modeling with Spreadsheets Professor Munson Topic 11 Sports Applications of Management Science

“Assigning Season Tickets Fairly” Problem: Friends purchase a block of Seattle Mariners season tickets for all 81 games and must divide them amongst themselves Solution: Mixed-integer programming (modification of the assignment problem)

Difficulties with Simple Rankings If want one game in a series, member might rank all three highly and then get all three! Games might not be spread out over the season Persons traveling from out of town might want the whole series Generally harder for the computer to solve than a binary program with many constraints

Some Extra Constraints No more than one game in any three-week stretch No more than one game per month Have the games spread out over the season No more than one game in each series No more than one game in each weekday series No more than two games on any weekend No more than one pair of tickets per home stand No more than one game against any given opponent No games in April A game with all four tickets in August At least two days between any two games Games with all four tickets should occur on Thursday, Friday, Saturday, or Sunday No games while on vacation No games with all four tickets on school nights Games for both Friday and Saturday or for neither

Notation for the Model Let n = number of participating buyers Let pi = number of games for which person i wanted 2 tickets Let qi = number of games for which person i wanted 4 tickets Let pij = 1 if person i is assigned 2 tickets to game j Let qij = 1 if person i is assigned 4 tickets to game j Let αij = weight that person i assigns to game j Let e = average satisfaction of the most dissatisfied person Let c = a large number (10,000)

The Formulation

Conclusions This approach effectively incorporated multiple objectives Solution time was unwieldy, but near-optimal solutions worked well Most tickets were assigned to people who really wanted to go Should there be limits/rules regarding personal constraints?

“Using Bivalent Integer Programming to Select Teams for Intercollegiate Women’s Gymnastics Competition” College gymnasts must be assigned to events according to the following rules: Four events: vault, uneven bars, balance beam, and floor exercise Each team can enter up to six gymnasts per event Top five scores included in the team score Must be at least four all-around participants

The Problem Coach must compute expected scores for different lineups This is an onerous combinatorial problem of nontrivial magnitude Competitions are often extremely close, so heuristics may not work well enough (1% from optimal may reduce score by 1.8 points) Potential heuristics?

Notation for the Model Let N be the number of team members Let Sij = expected score of gymnast i in event j Xij = 1 if gymnast i is a specialist in event j Yi = 1 if gymnast i is an all-arounder

The Formulation

Issues Program can be written as an LP by adding 0.0001 to the objective function coefficients of the Y variables What should Sij represent?

Other Issues Is there a better objective function? Program can easily be re-run to account for injuries Results are useful to explain to the participants themselves why the selections were made (e.g., must have 4 all-arounders) Model may provide alternate optima that could allow for rotations

“Baseball, Optimization, and the World Wide Web,” Playoff spots in major league baseball “Has my team clinched a playoff spot?” “Has my team been eliminated?” “How many wins does my team need to clinch a playoff spot?” “My team must win at least how many games to avoid elimination?” Problem: “Games back” and “magic number” are overly conservative and do not account for the remaining schedule Solution: Optimization

MLB Playoff System Each team plays 162 regular season games Each of two leagues (American and National) invites 4 teams: 3 Division winners plus the best second-place team (by record) Ties are broken by a one-game playoff Each league has a tournament to determine the pennant winner, and those two teams meet in the World Series of Major League Baseball

Elimination Questions When has a team been eliminated from the playoffs? Traditional approach: If a team trails the first-place team in wins by more games than it has remaining, it is eliminated Problem: Doesn’t consider the schedules of all other teams

Example: Detroit Tigers, 8/30/96 Team Wins Losses Games Back Games Left New York 75 59 — 28 Baltimore 71 63 4 Boston 69 66 6.5 27 Toronto 72 12.5 Detroit 49 86 26.5 Opponents Games Remaining Baltimore vs. Boston 2 Baltimore vs. New York 3 Baltimore vs. Toronto 7 Boston vs. New York 8 Boston vs. Toronto New York vs. Toronto

Hypothetical: Detroit wins final 27 games; N. Y Hypothetical: Detroit wins final 27 games; N.Y. beats Boston once and loses other 27 games; Boston beats N.Y. 7 times and loses other 20 games Team Wins Losses Games Back Detroit 76 86 — Boston New York Baltimore ? Toronto

Elimination “Numbers” (1) First-Place Elimination Number: Minimum number of remaining games that the team must win to have any chance of finishing in first place in the division (2) Play-Off Elimination Number: Minimum number of remaining games that the team must win to have any chance of earning a play-off spot (whether division or wild-card)

Clinch Questions When has a team clinched a play-off spot? Traditional approach for the first-place team: “Magic Number” = games remaining − (losses of 2nd place team − losses of 1st place team) Any combination of wins by the first-place team and losses by the second-place team totaling the magic number guarantees at least a tie for the top spot When the magic number drops to 0, the first-place team has won Problems: Only works for 1st place team and is not independent of other teams

Clinch “Numbers” (1) 1st Place Clinch Number: Minimum number of games which, if won, guarantees that the team finishes in at least a tie for first place (2) Play-Off Clinch Number: Minimum number of games which, if won, guarantees that the team has a position in the play-offs (either as a division winner or as the wild-card team)

Solution The clinch and elimination numbers are reported for each team every day during the baseball season A software system updates the standings nightly, runs the solution algorithms, and reports the results—all automatically Uses a free Internet news service, Infobeat, to automatically send an email message each night containing final scores of MLB games

As of Start of Play, Tuesday, April 20, 2010 http://lyle.smu.edu/~olinick/riot/baseball_main.html As of Start of Play, Tuesday, April 20, 2010

Notation xij = number of future games team i wins against team j gij = number of games remaining between teams i and j Dk = set of teams in Division k vk = Division k’s first-place-elimination threshhold wi = team i’s current number of wins

Formulation for 1st Place Elimination For any team i, vk − wi is its 1st place elimination #

Conclusions The article also provides optimization programs for two of the other “numbers.” The first-place clinch number can be found via the provided arithmetic calculations Developed and housed at Cal-Berkeley The home page has other educational links describing OR applications

“Scheduling the Chilean Soccer League by Integer Programming” Soccer is Chile’s most popular sport But in the early 2000s, fan support began to waiver Also, the current manual schedule lacked fairness, created poor travel conditions for teams and TV, and generated poor end-of-season matchups An integer programming model came to the rescue

Map of Chile

League Conditions in Chile Two Divisions: First (20 teams) and Second (12 teams) Two Seasons: Opening and Closing First Division is divided into 4 groups Three dominant teams: All of their games are shown on TV; plus 1-2 other TV games per week 19-week regular season 2 teams from each group make the playoffs

Objective Function Objective: Push “decisive” games towards the end of the season (feasibility more important than optimality) Let xijk = 1 if team i plays at home against team j in round k Let t(e) = set of teams in group e

Constraints Basic Scheduling Home and Away Sequence Each team must play each other team once Either 9 or 10 games must be at home Home and Away Sequence No more than two consecutive home games No more than two consecutive away games 1st 2 games must alternate home-away and last 4 games must alternate home-away

More Constraints Home Game Balance Against Group Rivals Two group opponents played at home and 2 away Geographic Constraints for Double Away Game Sequences No 2-game trips from North (South) to South (North) At least one game in own region

Even More Constraints Constraints on the Three Highly Popular Teams Must play either COLO at home and UCH away or vice-versa The 3 popular teams must play between weeks 10 and 16 Each of the three plays exactly one home game against the other two TV Constraints The 3 teams must play in Central or either North/South (but not North and South in same wk.) None plays away from home during first 5 rounds (summertime when mobile TV is less available)

Still More Constraints Constraints on Strong Teams No team may play 2 straight games against the set of four strongest teams Games between the 4th best team and the three best must be played between rounds 4 and 18 Crossed Teams Teams that share the same stadium cannot both be home during the same week Regional Classic Matchups Regional rivals must play between rounds 8 and 18

Can You Believe More Constraints? Santiago Games Santiago must host between 2 and 4 home games each week (there are 7 teams from Santiago) The four least popular Santiago teams should not play each other during the first five rounds (summertime) because attendance would be low Tourism-Related Constraints Each team in a tourist area (e.g., the beach) plays at home against at least one of the popular teams during the first five rounds (summertime)

Impacts 35% rise in attendance during the first year 74% increase in rival game attendance More than 100% revenue increase in certain regions TV stations saved money NO COMPLAINTS!!!