The National Basketball Association (NBA) is composed of 30 teams which are divided into two conferences of three divisions with five teams each. NBA games are divided into regular season and playoffs. The NBA regular season follows an asymmetrical round-robin tournament (otherwise known as uneven paired competition) format where the number of games between any two teams depends on their conference and divisional affiliations. In an NBA season each team plays 82 games, 41 at home and 41 on the road; in total, there are 1230 games in an NBA season. Each regular season takes approximately 180 days, spanning from mid-October to mid-April. Games are scheduled throughout the day, every single day except the All-star game break which is a one-week period in which the NBA All-star game as well as other events (e.g., slam dunk contest, three-point shooting contest, rising stars match, etc.) take place. Holidays (e.g., Christmas and MLK day) and weekends are more popular times to watch NBA games. Multiple national and local TV channels air the NBA games, including ESPN, TNT, ABC, and NBA-TV. Scheduling the entire 1230-game season and assigning games to dates, arenas, and TV channels is an extremely challenging task. Matt Winick, Senior Vice President of Scheduling and Game Operations is the main person responsible for producing the NBA regular season schedule. Matt has decided to break the scheduling task into two parts: i) for each four-day period, selecting a set of games that should be played, and then ii) assigning the selected games to specific days within this 4-day period. Matt's wishes to produce a schedule that maximizes the revenue generated from airing TV commercials. In this assignment, you will help Matt produce the NBA game schedule for a four-day time period which spans Thursday, February 4 through Sunday, February 7. The following table lists the games Matt has selected to be played during this time period, as well as popularity ratings for each game on a scale of 0 to 2, with 0 being the least popular and 2 being the most popular. The popularity rating is determined by the ranking of the host and guest teams, location, and overall popularity of those franchises. Matt estimates that the NBA's revenue for each game is directly related to its popularity rating. In general, games played on Thursday generate $1 million in revenue for each popularity point, Friday games generate $1.5 million in revenue for each popularity point, and weekend games generate $2 million in revenue for each popularity point. However, since Sunday, February 7th coincides with the National Football League (NFL) Super Bowl, the games played on that Sunday will generate only $0.5 million in revenue for each popularity point. As an example, the first game (Celtics at Lakers) in the table below would generate $2M in revenue if played on Thursday, $3M if played on Friday, $4M if played on Saturday, and $1M if played on Sunday. Game Abbreviation Popularity Rating 1 Boston Celtics @ LA Lakers BOS - LAL 2 2 Atlanta Hawks @ LA Clippers ATL - LAC 0.75 3 Portland Trail Blazers @ Milwaukee Bucks POR - MIL 1.25 4 Denver Nuggets @ Washington Wizards DEN WAS 0.6 5 Utah Jazz @ Brooklyn Nets UTA-BKN 1.5 6 Indiana Pacers @ Dallas Mavericks IND-DAL 1.1 7 Charlotte Hornets @ New Orleans Pelicans CHA-NOP 1.2 8 New York Knicks @ Houston Rockets NYK - HOU 0.6 9 Toronto Raptors @ Phoenix Suns TOR - PHX 1 10 Chicago Bulls @ Cleveland Cavaliers CHI - CLE 0.45 11 Miami Heat @ Orlando Magic MIA - ORL 1.1 12 Detroit Pistons @ San Antonio Spurs DET - SAS 0.5 13 Sacramento Kings @ Golden State Warriors SAC - GSW 1.25 14 Memphis Grizzlies @ Oklahoma City Thunder MEM-OKC 0.75 15 Philadelphia 76ers @ Minnesota Timberwolves PHI - MIN 1.3 16 Toronto Raptors @ LA Lakers TOR - LAL 1.9 17 Phoenix Suns @ Golden State Warriors PHX - GSW 1.4 18 LA Clippers @ Sacramento Kings LAC-SAC 0.9 19 Utah Jazz @ Washington Wizards UTA-WAS 1.1 20 New York Knicks @ San Antonio Spurs NYK - SAS 0.8 The number of game slots for each day during this four-day period is given in the table below: Date Game slots available Thursday, February 4 5 Friday, February 5 6 Saturday, February 6 6 Sunday, February 7 3 Matt must consider the following managerial constraints when producing the schedule: i. Each team should play at most one game each day. ii. At least one game with a popularity rating of 1.25 or higher should be assigned to each day. iii. Due to using the same arena (i.e., Staples Center) the two Los Angeles teams, Lakers and Clippers, cannot host a game on the same day. iv. Because of Super Bowl, Sunday cannot have a game with popularity rating more than 1.4. Continuing their road trip from the days prior to this 4-day block, the New York Knicks and Utah Jazz will need to play the rest of their road games as back-to-back games on Thursday and Friday, with no games on Saturday and Sunday to have time to get back to their home cities for their Monday home games. V. Given the above information, please answer the following questions: 1. Help Matt formulate the problem of scheduling games during the four-day period while maximizing revenue. Explicitly write down your decision variables, the objective function, and constraints. What kind of optimization problem have you defined? The National Basketball Association (NBA) is composed of 30 teams which are divided into two conferences of three divisions with five teams each. NBA games are divided into regular season and playoffs. The NBA regular season follows an asymmetrical round-robin tournament (otherwise known as uneven paired competition) format where the number of games between any two teams depends on their conference and divisional affiliations. In an NBA season each team plays 82 games, 41 at home and 41 on the road; in total, there are 1230 games in an NBA season. Each regular season takes approximately 180 days, spanning from mid-October to mid-April. Games are scheduled throughout the day, every single day except the All-star game break which is a one-week period in which the NBA All-star game as well as other events (e.g., slam dunk contest, three-point shooting contest, rising stars match, etc.) take place. Holidays (e.g., Christmas and MLK day) and weekends are more popular times to watch NBA games. Multiple national and local TV channels air the NBA games, including ESPN, TNT, ABC, and NBA-TV. Scheduling the entire 1230-game season and assigning games to dates, arenas, and TV channels is an extremely challenging task. Matt Winick, Senior Vice President of Scheduling and Game Operations is the main person responsible for producing the NBA regular season schedule. Matt has decided to break the scheduling task into two parts: i) for each four-day period, selecting a set of games that should be played, and then ii) assigning the selected games to specific days within this 4-day period. Matt's wishes to produce a schedule that maximizes the revenue generated from airing TV commercials. In this assignment, you will help Matt produce the NBA game schedule for a four-day time period which spans Thursday, February 4 through Sunday, February 7. The following table lists the games Matt has selected to be played during this time period, as well as popularity ratings for each game on a scale of 0 to 2, with 0 being the least popular and 2 being the most popular. The popularity rating is determined by the ranking of the host and guest teams, location, and overall popularity of those franchises. Matt estimates that the NBA's revenue for each game is directly related to its popularity rating. In general, games played on Thursday generate $1 million in revenue for each popularity point, Friday games generate $1.5 million in revenue for each popularity point, and weekend games generate $2 million in revenue for each popularity point. However, since Sunday, February 7th coincides with the National Football League (NFL) Super Bowl, the games played on that Sunday will generate only $0.5 million in revenue for each popularity point. As an example, the first game (Celtics at Lakers) in the table below would generate $2M in revenue if played on Thursday, $3M if played on Friday, $4M if played on Saturday, and $1M if played on Sunday. Game Abbreviation Popularity Rating 1 Boston Celtics @ LA Lakers BOS - LAL 2 2 Atlanta Hawks @ LA Clippers ATL - LAC 0.75 3 Portland Trail Blazers @ Milwaukee Bucks POR - MIL 1.25 4 Denver Nuggets @ Washington Wizards DEN WAS 0.6 5 Utah Jazz @ Brooklyn Nets UTA-BKN 1.5 6 Indiana Pacers @ Dallas Mavericks IND-DAL 1.1 7 Charlotte Hornets @ New Orleans Pelicans CHA-NOP 1.2 8 New York Knicks @ Houston Rockets NYK - HOU 0.6 9 Toronto Raptors @ Phoenix Suns TOR - PHX 1 10 Chicago Bulls @ Cleveland Cavaliers CHI - CLE 0.45 11 Miami Heat @ Orlando Magic MIA - ORL 1.1 12 Detroit Pistons @ San Antonio Spurs DET - SAS 0.5 13 Sacramento Kings @ Golden State Warriors SAC - GSW 1.25 14 Memphis Grizzlies @ Oklahoma City Thunder MEM-OKC 0.75 15 Philadelphia 76ers @ Minnesota Timberwolves PHI - MIN 1.3 16 Toronto Raptors @ LA Lakers TOR - LAL 1.9 17 Phoenix Suns @ Golden State Warriors PHX - GSW 1.4 18 LA Clippers @ Sacramento Kings LAC-SAC 0.9 19 Utah Jazz @ Washington Wizards UTA-WAS 1.1 20 New York Knicks @ San Antonio Spurs NYK - SAS 0.8 The number of game slots for each day during this four-day period is given in the table below: Date Game slots available Thursday, February 4 5 Friday, February 5 6 Saturday, February 6 6 Sunday, February 7 3 Matt must consider the following managerial constraints when producing the schedule: i. Each team should play at most one game each day. ii. At least one game with a popularity rating of 1.25 or higher should be assigned to each day. iii. Due to using the same arena (i.e., Staples Center) the two Los Angeles teams, Lakers and Clippers, cannot host a game on the same day. iv. Because of Super Bowl, Sunday cannot have a game with popularity rating more than 1.4. Continuing their road trip from the days prior to this 4-day block, the New York Knicks and Utah Jazz will need to play the rest of their road games as back-to-back games on Thursday and Friday, with no games on Saturday and Sunday to have time to get back to their home cities for their Monday home games. V. Given the above information, please answer the following questions: 1. Help Matt formulate the problem of scheduling games during the four-day period while maximizing revenue. Explicitly write down your decision variables, the objective function, and constraints. What kind of optimization problem have you defined