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Hw5s - ENGRI 1101 Engineering Applications of OR Fall 2008...

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ENGRI 1101 Engineering Applications of OR Fall 2008 Homework 5 Homework 5 Solutions 1. Thank you for your comments! 2. The Giants cannot win (or tie for the win). If they win all of their remaining games, they will end the season with 59 + 22 = 81 total wins. Notice that, for the Giants to win, neither the Dodgers nor the Padres can win more than 3 of their remaining games. However, since these two teams still have to play 7 games with each other, one of them is going to earn at least 4 more wins, eliminating the Giants. 3. (a) The hockey (or soccer) elimination problem can be viewed as a more general baseball elimination problem since it allows for ties to happen. Instead of the points for a match going to just one of the competing teams, there is the possibility that they’re split evenly. Still, and this is the crucial point you should have understood, each game is worth two points, no matter how it ends, and these two points either go to the winner or one to each team in case of a tie. There are several ways to formulate this precisely, we’ll give one possibility. The input we need for this problem consists of the current standings, expressed as the number of points each team has accumulated so far; call them p ( i ) (and the unit is points here!) for teams i = 1 , 2 , . . . , n and the number of games g ( i, j ) to be played between each of the possible pairings { i, j } . The output, what we want to know as a solution to this problem, is how each game should end (or at least one combination of possibilities of how each game should end) so that our team n is not eliminated. That is, we want to know how many points y ( i, j ) each team i gets from the games with each team j . If our team n is already eliminated, we want a proof, a convincing explanation. As said before it is crucial to realize that there are two points to be distributed amongst the teams in each games, that’s why the restriction y ( i, j ) + y ( j, i ) = 2 g ( i, j ) = 2 g ( j, i ) for all i, j ∈ { 1 , 2 , . . . , n - 1 } (1) must hold for the solution.

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