Question

You are a couch of the national basketball team and should select 12 players for the upcoming international tournament. There are 20 candidate players, p1,...,p20. For each player, you have collected several statistics that can be summarized as follows: For player i, the player’s rebounding average ri, assists average ai, height hi, scoring average si, and overall defense ability di. The players have been divided into four categories: play makers (PM) (p1,...,p5), shooting guards (SG) (p4,...,p11), forwards (F) (p9,...,p16), and centers (C) (p16,...,p20). Notice that some players can be used in multiple roles. Players p1,p8,p15,p20 are college students, while all other players are professional players. Answer the following questions.

(a) (5 points) Define decision variables.

(b) (5 points) You want to maximize the scoring average. Define objective function.

(c) (5 points) The team should consist of at least 3 PMs, 4 SGs, 4 Fs, 3Cs. Define constraints for this.

(d) (5 points) At least 2 players should be college students. Define constraints for this.

(e) (5 points) Mean rebounding, assists, scoring average, height, and defense ability should be at least r,a,s,h,d, respectively. Define constraints for this.

(f) (5 points) Player p5 has declared that if player p9 is selected, he does not want to be in the team. Define constraints for this.

(g) (5 points) Player p2 and player p19 can only be selected together because they have been played long for the same team. Define constraints for this.

(h) (5 points) Player p1,p7,p12,p17 are playing for the same team. You cannot select more than 3 players from the same team. Define constraints for this.

Answer 1

a) The decision variables are:

   $X_{i}=\left\{\begin{matrix} 1\ if\ the\ player\ i\ is\ selected\\ 0\ if\ not \end{matrix}\right.\ \forall i=1,2,...,20.$

b)

$O.F.:\ max\ z= \sum_{i=1}^{20}s_{i}X_{i}$

c)

$\sum_{i=1}^{5}X_{i}\geq 3\ (PM)$

$\sum_{i=4}^{11}X_{i}\geq 4\ (SG)$

$\sum_{i=9}^{16}X_{i}\geq 4\ (F)$

$\sum_{i=16}^{20}X_{i}\geq 3\ (C)$

d)

$X_{1}+X_{8}+X_{15}+X_{20}\geq 2$

e)

$\sum_{i=1}^{20}X_{i}r_{i}\geq r\sum_{i=1}^{20}X_{i}\(Mean\ rebounding)$

$\sum_{i=1}^{20}X_{i}a_{i}\geq a\sum_{i=1}^{20}X_{i}\(Mean\ assists)$

$\sum_{i=1}^{20}X_{i}h_{i}\geq h\sum_{i=1}^{20}X_{i}\(Mean\ height)$

$\sum_{i=1}^{20}X_{i}s_{i}\geq s\sum_{i=1}^{20}X_{i}\(Mean\ scoring)$

$\sum_{i=1}^{20}X_{i}d_{i}\geq d\sum_{i=1}^{20}X_{i}\(Mean\ defense)$

f)

$X_{5}\leq 1-X_{9}$

g)

$X_{2}=X_{19}$

h)

$X_{1}+X_{7}+X_{12}+X_{17}\leq 3$

Commentary: If I have to select 12 players out of 20, I should write:

$\sum_{i=1}^{20}X_{i}=12$