Question

After the bad start to the season and unexpected injuries, the Dallas Cowboys wants to sign 3 new players. There are five players under consideration. Player 1, 2 and 4 can play the quarterback position and Cowboys want to bring in at least one new player for this position. Each player has a strength given by sj, and Cowboys want to add at least S units of strength to the team by hiring some of these 5 players. On the other hand, the Cowboys will be careful while signing people who are prone to be penalties. Player j is likely to have at. penalties per season and Cowboys impose a quota of A penalties per season for these players. (a) If player j demands the salary cj, make up an IP to decide on which players should be signed to minimize the signing budget. (Just write out the IP) (b) If players 1 and 2 are buddies and will come to Dallas only together, add a constraint to guarantee that either they are signed together or they are not signed at all. (Just write this new constraint) (c) If player 1 hates player 3 and will not come if player 3 comes, add an appropriate constraint that does not allow signing player 1 and 3 together. (Just write this constraint)

Answer 1

Answer:

Assume the player 1 be denoted by j1, player 2 by j2, player 3 by j3, player 4 by j4, player 4 by j5.

If player j demands the salary cj, then the IP can be formulated as follows:

Objective:

Minimize:

cj1+cj2+cj3+cj4+cj5

Subject to:

1. cj1 = cj2 (Player 1 and 2 shoukd be signed together)

2. j1+j2+j3+j4+j5 =3 (3 players should be signed from 5 players)

3. j1+j2+j4 >=1 (Atleast 1 player must be chosen from player 1,2 and 4)

4.sj1+sj2+sj3+sj4+sj5 >= S units (Strengths of the player must be S units)

5. aj1+aj2+aj3+aj4+aj5 <= A quota (Penalties must be less than A quota)

6. j1<j3 or j3<Ji (Player 1 and 3 will not come together)

7. j1,j2,j3,j4,j5 >=0 (Non-negative constratint)