T HE D RAPERTON P ARKS AND R ECREATION D EPARTMENT'S F ORMATION OF G IRLS' B ASKETBALL T EAMS
Each fall the Draperton Parks and Recreation Department holds a series of tryouts for its boys' and girls' youth basketball leagues. All leagues are formed by age group, each usually encompassing a couple of ages. One such grouping is the 12- to 13-year-old girls' league. This is a particularly competitive age group because the girls are all trying to improve their skills in hopes of making the high school junior varsity team in the immediate future and the high school varsity in several years . Some of the girls even hope to make the high school varsity next year, when they are freshmen. This is also the first age group in which not all girls who try out are placed on teams ; some are cut. In the younger age groups, all players are assigned to teams. However, at this age group, four teams are formed, each with only seven players. This policy was agreed upon by a committee consisting of the Recreation Department director, a group of parents of past players, and the high school boys' and girls' basketball coaching staff. The small roster size allows each player to get a lot of playing time and makes the games more competitive. It also makes the teams more competitive in tournaments with teams from other towns and cities. This makes the girls very competitive, and it makes their parents even more competitive. Sandy Duncan, the Recreation Department's director of team sports, knows from past experience that parents do a lot of jockeying and politicking during tryouts. This has given rise to parental protests regarding the tryout process and about how players are selected and assigned to teams. Sandy and the player evaluators have frequently been accused of showing favoritism and stupidity.
This year, in an effort to blunt some of this criticism, Sandy has devised a tryout process that will hopefully remove her (and any other person) from direct involvement with the player selection and team formation process. First, she conducted tryouts for all 36 players who were trying out over two weekends that included four sessions each weekend : one on Friday evening, two on Saturday, and one on Sunday afternoon. She formed an evaluation team of several coaches from other towns and some of the current players and assistant coaches from the nearby State University women's basketball team. Each player was assigned a tryout number and evaluated for different skills and ability, and at the end of the tryouts, each player was given an overall score from 1 to 10, with 10 being the highest, as follows :
Sandy took a course in management science while she was a student at State University, and she wants to develop an integer linear programming model to use these evaluation scores to select the players and assign them to the four teams in a fair and equitable manner. She wants to select the best players, and in order to have competitive, equally matched teams, she wants each team to have an average overall player evaluation score between 6 and 7. As she sat down one evening to work on this problem, she quickly realized that it would require a relatively complex integer programming model. Help Sandy formulate and solve an integer programming model for this problem to select the best players, assign seven players to each team, and achieve the average team evaluation score Sandy wants for each team. Compare the teams formed by your model and determine whether you think they are competitive (i.e., whether the model has achieved Sandy's objectives). Also, determine whether any deserving players were unfairly cut by the model.