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Practice Problems |
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Practice Problems |
Easy Round Robin Scheduling Practice Problem
Hey guys, its scheduling time! You have decided to run a small co-recreational basketball league for faculty and staff members at your university during the traditional faculty/staff lunch times. You have reserved your facility, the Turner Center, Monday October 12 through Friday, October 16 for 1 week. You have 10 teams, which have signed up for your league (Nets, Knicks, Bulls, Kings, Pacers, Magic, Pistons, Wolves, Mavs and Rockets). You have 2 courts reserved to play games from 12:00 p.m. to 2:00 p.m. You have guaranteed each team will play at least 3 games in the tournament. Games will last 1 hour and teams should not play more than one game per day. You are offering only a co-recreational tournament...no other units of participation will be offered (skill level, etc.).
To complete the schedule, provide the following information
· Draw a master facility schedule for both courts, showing available game times and the leagues that will play at those times.· Break teams down into an appropriate number of leagues, making sure that the total number of games it takes to play the tournament does not exceed the total number of games available to you.
· Create a schedule for the teams in both leagues, depicting their opponents, days, times and playing sites for all of the rounds in their tournament.
· Tell me how many games each TEAM will play in the tournament.
· Tell me how many games it will take to complete each LEAGUE.
Practice ProblemMedium Difficulty Round Robin Scheduling
You have been contracted to run an AAU basketball tournament for high school teams in the state of South Dakota. You have ample amount of facility space reserved to run your tournament. You are playing games at the HPER Building at SDSU, and you have secured the following facility reservation for your tournament: you have 6 courts available (okay pretend there are 6) to you between the hours of 1:00 p.m. and 4:00 p.m. You have decided to play games on the hour. Games will be played at 1:00 p.m., 2:00 p.m. and 3:00 p.m. on all 6 courts. You have the HPER reserved for 5 days, beginning with Monday and ending with Friday of the same week.
You had a really positive registration response, as you had 18 men's teams and 18 women's teams (total of 36 teams) sign up to play. All of the players are of basically the same skill level, so you are only going to break leagues up by gender (no other units of participation). Teams shouldn't play more than once per day. The teams that have entered your tournament are:
Men's and Women's Teams represent the following AAU programs:
Brookings, Mitchell, Madison, Souix Falls Washington, Watertown, Aberdeen, Huron, Souix Falls Roosevelt, Souix Falls Lincoln, Vermillion, Volga, Old Ham-Romana, Milbank, Pierre, Spearfish, Rapid City, and Baltic
To complete the schedule, provide the following information
· Draw a master facility schedule for both weeks, showing available game times and the leagues that will play at those times.· Break teams down into an appropriate number of leagues, making sure that the total number of games it takes to play the tournament does not exceed the total number of games available to you.
· Create a schedule for the teams in both leagues, depicting their opponents, days, times and playing sites for all of the rounds in their tournament.
· Tell me how many games each TEAM will play in the tournament.
· Tell me how many games it will take to complete each LEAGUE.
Difficult Round Robin Scheduling Practice Problem
A professional minor league ice hockey franchise has relocated to Brookings, South Dakota and you have been hired as their first ever Director of Community Relations. One of the concerns of the new ownership group is that they are worried about the potential fan base for ice hockey in mid-South Dakota, and want to create some excitement about their new team and the upcoming season. As such, you have been given the task of organizing an adult ice hockey tournament that will play its games at the new hockey team's practice facility. The pro team's players will serve as coaches and the players will all receive pro-style hockey gear free of charge from the team.
For your first tournament, registration response has been good. You had 17 men's teams register to play. Unfortunately, no women's teams were interested this year. Of the 17 men's teams, 5 teams were considered "advanced" level, 8 teams were considered "intermediate" level and 4 teams were considered "beginner" level.
You have been able to reserve the ice hockey arena for 4 total days, beginning Monday, March 1st and ending Thursday, March 4. You have a total of 2 ice hockey rinks available to you, and you have decided to play games on the hour, at 4pm, 5pm, 6pm, and 7pm on both rinks.
The "advanced" level teams are: Devils, Rangers, Islanders, Flyers, Red Wings
The "intermediate" level teams are:
Bruins, Sabres, Blackhawks, Canadiens, Leafs, Blues, Stars, PanthersThe "beginner" level teams are: Kings, Coyotes, Sharks, Flames
To complete the schedule, provide the following information
· Draw a master facility schedule for the week, showing available game times and the leagues that will play at those times.· Break teams down into an appropriate number of leagues, making sure that the total number of games it takes to play the tournament does not exceed the total number of games available to you.
· Pick any ONE league: create a schedule for the teams in that league depicting their opponents, days, times and playing sites for all of the rounds in their league.
· Tell me how many games each TEAM will play in the tournament.
· Tell me how many games it will take to complete each LEAGUE.
A) Odd number of teams per league:Under the conditions stated below, find the NUMBER OF WEEKS AND THE DAY OF THE WEEK in which a round robin tournament should end.Round Robin Scheduling
Type I:
Forecasting Days
Problem #1ASSUMPTIONS: Regularly scheduled games are played Monday through Thursday. Postponed games can be completed on weekends. Two days are set aside at the end of league play to cover the possibility of league ties.
N = 35 teams. There are five leagues of 7 teams. The first day of play is Wednesday and the number of games played per day is 8.
Step 1. Determine the maximum number of games that can be scheduled per day without having a team play twice in the same day. To do this, you must find the number of games per round for each of the leagues and add them together. Once this is completed, the total number of games per round for all leagues must be checked to make sure that it does not exceed the number of games available per day. (Note: if the total number of games per round is less than the number of games available per day, an adjustment is necessary in order to prevent a team playing more than one game per day. In other words, the number of games per round would be reduced to equal the number of games available per day)
Use the formula for Odd number of teams per league: N = # of teams
(N-1) = ( -1) = _____ = games per league X leagues = games per round
2 2 2Since only 8 games are available per day, the number of games per round (15) exceeds the number of games that could be scheduled in one day, so no adjustment is needed. It will take almost 2 days before teams repeat.
Step 2. Determine the total number of league games.
N(N-1) = ( -1) = ( ) = = games per league X leagues
2 2 2 2= total games
Step 3. Determine the total number of days it will take to complete the tournament. Divide the total number of league games ( ) by the number of games that are available each day ( ).
/ = days or days. (Always round up)
Step 4. Since we must add two days to the tournament scheduling process to take care of league ties, rainouts, etc., we add 2 to the total number of days figured in Step 3.
days + days = days are needed to schedule the tournament.
Step 5. The last step is to determine the actual number of playing weeks and the ending day of the tournament. To accomplish this, divide the total number of tournament days ( ) by the number of days scheduled per week ( days - Monday through Thursday). Remember the starting day of the tournament was a Wednesday.
days / days per playing week
M T W R Week 1 (2 days)
M T W R Week 2 (6 days)
M T W R Week 3 (10 days)
M T W R Week 4 (14 days)
M T W R Week 5 (16 days)Answer: The tournament will be completed on of the 4th playing week or th calendar week.
Round Robin Scheduling
Type II:
Minimum Number of Games
Problem #2Under the conditions stated below, find the MINIMUM NUMBER OF GAMES that must be available for each day to complete a round robin tournament.
ASSUMPTIONS: Regularly scheduled games are played Monday through Thursday. Postponed games can be completed on weekends. Two days are set aside at the end of league play to cover the possibility of league ties, rainouts, etc.
N = 34 teams. There are six leagues of 5 teams and 1 league of 4 teams. Four weeks are available for play.
Step 1. Determine the total number of league games. N= # of teams. You need to do the problem for each league and then add them together for total league games.
A) N(N-1) = ( -1) = ( ) = = games per league
2 2 2 2X leagues = total games
B) N (N-1) = ( -1) = ( ) = = games per league
2 2 2 2X league = total games
Therefore, games + games = total league games
Step 2. Determine the total number of days that are available to conduct the tournament using the number of weeks and the number of days per week given.
days per week (Mon, Tues, Wed and Thurs) X weeks = ______ days available for play
Step 3. Since two days are set aside at the end of league play to cover the possibility of league ties, rainouts, etc. these two days must be deducted from the total number of days determined in Step 2.
days - days for league ties, rainouts, etc. = total adjusted days available for play
Step 4. Determine the minimum number of games that must be available per day. To find this figure, divide the total number of league games ( ) by the total adjusted days available for play ( ).
/ = or games (always round up) must be available per day in order to complete the tournament in the given time span of five weeks.
Step 5. Check to make sure that teams will not play more than one game per day. In other words, the number of games available per day (Step 4) must be less than or equal to the maximum total number of games per round. We must determine the maximum number of games per round that can be scheduled before proceeding into round two of the tournament. To figure the total number of games per round, use the following formulas: N = # of teams
N - 1 = - 1 = = games/round
2 2 2leagues x games/round = games/rd
B) Even number of teams per leagues: N = = games/round
2 2
league x games/round = games/rdTherefore, Total number of games per round = + = 14 total games per round
Since our calculation of five games per day (Step 4) is less than the total number of games per round (14), teams will not play more than once per day. (It will take almost 3 days for all teams to play at least once.) So our answer of five games per day is correct.
Answer: A minimum of five games must be available per day in order to complete the tournament.
Round Robin Scheduling
Type III:
Maximum Number of Entries
Problem #3Under the conditions stated below, find the MAXIMUM NUMBER OF ENTRIES that can be scheduled in a round robin tournament.
ASSUMPTIONS: Regularly scheduled games are played Monday through Thursday. Postponed games can be completed on weekends. Two days are set aside at the end of league play to cover the possibility of league ties.
Four weeks are available and 5 games can be played per day.
Step 1. Determine the actual number of days that are available for play. This includes days set aside for league ties, rainouts, etc.
Total days available = Days available - (League ties, rainouts, etc.)
= ( weeks X days per week) - ( days set aside)
= - = total days available
Step 2. Determine the total number of league games possible.
Total league games = Total days available X Number of games available per day
= X = total league games possible
Step 3. Determine the number of games required to complete a 3-team round robin. N=# of teams
N(N - 1) = ( - 1) = ( ) = = games per league
2 2 2 2Step 4. Determine the number of leagues possible by dividing the number of games required per league (3) into the total number of league games possible (70).
/ = or leagues (always round down)
Step 5. Determine the total number of teams that can be scheduled. This is calculated by multiplying the number of leagues ( ) times the number of entries in each league ( ).
leagues X teams per league = total teams
ANSWER: 69 total teams can be accommodated in the tournament.