The National Basketball Dream Team
Rudy Dellamico has been the coach of the United States national men's basketball team for the last two years. The national men's basketball team is one of the 16 finalists for the world championship basketball tournament, which is scheduled to take place in Barcelona next summer. Rudy is the chair of the player selection committee, which is responsible for selecting the 12 players that will comprise the ''dream team" from among the best players in the National Basketball Association (NBA) and the National Collegiate Athletic Association (NCAA) men's basketball league. The committee has recently narrowed the field of possible dream team players to 20 players whose player statistics are shown in the spreadsheet DREAM-TEAM.PDF.
The players have been divided into four play positions:
• Players 1, 2, 3, 4, and 5 are designated as "playmakers", otherwise known as pointguards.
• Players 4, 5, 6, 7, 8, 9, 10, and 11 are designated as "shooting guards."
• Players 9, 10, 11, 12, 13, 14, 15, and 16 are "forwards."
• Players 16, 17, 18, 19, and 20 are "centers."
Notice that there are players that can play more than one position. For example, player 4 is both a playmaker and shooting guard. For balance purposes, the committee has decided that the team should consist of at least three playmakers, four shooting guards, four forwards, and three centers, which implies that some players who play multiple positions need to be selected.
Players 4, 8, 15, and 20 play in the NCAA, while all of the other players play in the NBA. The committee has decided that they would like at least two players from the NCAA to be included on the dream team.
After a lot of wrangling, the committee has decided that the average rebound ability among the 12 selected players should be at least 7 rebounds per game. Also, they have decided that the average number of assists among the 12 selected players should be at least 6 assists per game. Furthermore, they have decided that the average points scored among the 12 selected players should be at least 18 points per game. Finally, the committee has decided that the average height of the players should be at least 6'7'', and that the average defensive ability among the 12 selected players should be at least 8.5.
As if the selection problem were not already sufficiently complicated, alas there are compatibility problems among these highly-compensated athletic stars! Player 5 has declared that if player 9 is selected, then he will turn down any invitation to join the team. Also, players 2 and 19 can only be selected as a pair, since they have played on the same team for seven years and they
feel that they are much more effective when they play together.
Finally, in order for the committee not to be accused of undue favoritism, at most three players from the same franchise team can be selected. Only one franchise team has more than three players among the list of 20 players: Players 1, 7,
12, and 16 all play for the same team, the Boston Celtics.
Suppose that Rudy Dellamico would like to apply some real management science to the selection of the players for the dream team, and that his objective is to select the team with the highest average points scored per game subject to all of the requirements indicated above.
Assignment:
a) The spreadsheet DREAM-TEAM.PDF contains all of the data about the statistics of players. Using this spreadsheet, construct and solve a discrete optimization model that can be used to select the national men's basketball dream team.
b) How will the composition of the dream team be changed if the following constraints are changed as follows:
- the average number of rebounds among the 12 selected players needs to be at least 8 rebounds per game,
- the average number of assists per game among the 12 selected players must be at least 7 assists per game, and
- the average number of points per game among the 12 selected players must be at least 20 points per game.
c) Rudy is thinking of also selecting two more players as alternates in case any of the 12 selected players is injured prior to the start of the world championship games in Barcelona. Propose a method for selecting the two alternates.
Attachment:- dream team.pdf