A football directortriesto form a five playersoccer team. There are seven player

Question

A football directortriesto form a five playersoccer team. There are seven players who have been graded (on a scale of 1: weak to 3: perfect)according to their ability 1, ability 2, ability 3, and ability 4. The positions that each player is permitted to play and the abilities of each player are shown in the below table. The formed five player team must satisfy the following constraints:

At least 4 playersmust be able to play at position 1, at least 2 playersmust be able to play at position 2, and at least 1 player must be able to play at position 3. The average ability 1, ability 2, and ability 3 of the team must be at least 2. If player 3 starts, then player 4 cannot start. If player 1 starts, then players 6 and 5 must both start. Either player 2 or player 3 must start. Given these constraints, the directorwants to maximize the total ability 4of the team. Formulate an IP for this

player position ability1 ability 2 ability 3 ability 4 1 1 3 3 1 3 2 3 2 1 3 2 3 1-2 2 3 2 2 4 2-3 1 3 3 1 5 1-2 3 3 3 3 6 2-3 3 1 2 3 7 1-2 3 2 2 1

Explanation / Answer

Give that atleast 4 players had to play at position one but we have only 4 players who can play at position 1 and of them 3 can play at position 2 which satisfies the first two constraints. thus players 1,3,5,7 are into the team

Now we had to choos the 5th player who can play at position3 and also the  average ability 1, ability 2, and ability 3 of the team must be at least 2.

So we have players 2,4 and 6 of which we had to choose one. But given that if player 3 starts, then player 4 cannot start.

So we are left with players 2,6.

Since if player 1 starts, then players 6 and 5 must both start and either player 2 or player 3 must start.

We have to choose player 6 as the 5th player of the team.

So the team contains players 1,3,5,6,7.

Thus the average of abilities1,2,3 = [(3+3+1)/3 + (2+3+2)/3 + (3+1+2)/3 + (3+1+2)/3 + (3+2+2)/3 ]/5 = [7/3 + 7/3 + 2+2+7/3]/5 = 11/5 =2.20

Thus this satifies the constraint that the  average ability 1, ability 2, and ability 3 of the team must be at least 2.

Now ability 4 of the team = 3+2+3+3+1= 12.

Thus the team contains players 1,3,5,6,7.

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