Exercise 10.2 (Selection of the dream team)

The coach of the national basketball team is faced with the decision of selecting 12 players for the upcoming international tournament. He has limited his final selection to 20 players, $p_{1}, \dots, p_{20}$ . For each player, the coach has collected several statistics that can be summarized as follows. His rebounding average $r_{i}$ , his assists average $a_{i}$ , his height $h_{i}$ , his scoring average $s_{i}$ , and his overall defense ability $d_{i}$ . The players have been divided into four broad categories: play makers (PM) $(p_{1}, \dots, p_{5})$ , shooting guards (SG) $(p_{4}, \dots, p_{11})$ , forwards (F) $(p_{9}, \dots, p_{16})$ , and centers (C) $(p_{16}, \dots, p_{20})$ . Notice that there are players that can be used in multiple roles (for example player $p_{4}$ can be used both as a play maker and a shooting guard). Players $p_{4}, p_{8}, p_{15}, p_{20}$ play in the NCAA (college level), while all of the rest play in the NBA (professional level). For balance purposes, the team should consist of at least 3 play makers, 4 shooting guards, 4 forwards, and 3 centers, which implies that some players with dual roles should be selected. In addition, at least 2 players from the NCAA should be selected, while the mean rebounding, assists, scoring average, height, and defense ability should be at least $\bar{r}, ā, \bar{s}, \bar{h}, \bar{d}$ , respectively. The problem is further complicated by the fact that there are compatibility problems among some of the players. Player $p_{5}$ has declared that if player $p_{9}$ is selected, then he does not want to be in the team. Also, players $p_{2}$ and $p_{19}$ can only be selected together as they play in the same team for years and feel that they are much more effective together. Finally, at most 3 players from the same team should be selected, so that the coach is not accused of favoritism (players $p_{1}, p_{7}, p_{12}, p_{16}$ play for the same team). Faced with these difficulties, the coach has decided that he would like to maximize the scoring average, while satisfying the various constraints. Formulate the problem that the coach is facing as an integer programming problem.

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The coach of the national basketball team is faced with the decision of selecting 12 players for the upcoming international tournament. He has limited his final selection to 20 players, $p_1, \ldots, p_{20}$. For each player, the coach has collected several statistics that can be summarized as follows. His rebounding average $r_i$, his assists average $a_i$, his height $h_i$, his scoring average $s_i$, and his overall defense ability $d_i$. The players have been divided into four broad categories: play makers (PM) $\left(p_1, \ldots, p_5\right)$, shooting guards (SG) $\left(p_4, \ldots, p_{11}\right)$, forwards (F) $\left(p_9, \ldots, p_{16}\right)$, and centers (C) $\left(p_{16}, \ldots, p_{20}\right)$. Notice that there are players that can be used in multiple roles (for example player $p_4$ can be used both as a play maker and a shooting guard). Players $p_4, p_8, p_{15}, p_{20}$ play in the NCAA (college level), while all of the rest play in the NBA (professional level). For balance purposes, the team should consist of at least 3 play makers, 4 shooting guards, 4 forwards, and 3 centers, which implies that some players with dual roles should be selected. In addition, at least 2 players from the NCAA should be selected, while the mean rebounding, assists, scoring average, height, and defense ability should be at least $\bar{r}, \bar{a}, \bar{s}, \bar{h}, \bar{d}$, respectively. The problem is further complicated by the fact that there are compatibility problems among some of the players. Player $p_5$ has declared that if player $p_9$ is selected, then he does not want to be in the team. Also, players $p_2$ and $p_{19}$ can only be selected together as they play in the same team for years and feel that they are much more effective together. Finally, at most 3 players from the same team should be selected, so that the coach is not accused of favoritism (players $p_1, p_7, p_{12}, p_{16}$ play for the same team). Faced with these difficulties, the coach has decided that he would like to maximize the scoring average, while satisfying the various constraints. Formulate the problem that the coach is facing as an integer programming problem.

Exercise 10.2 (Selection of the dream team)

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