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Unformatted text preview: ENGRI 1101 Engineering Applications of OR Fall 2008 Handout 6 The baseball elimination problem We shall now explore a much more sophisticated application of the maximum flow problem. While this application is traditionally phrased in terms of baseball, it really has nothing to do with that sport in particular, especially as it is now played in the major leagues. The setting is as follows. There are n teams: team 1, team 2, ..., team n . When two teams play each other in a game, one team wins, and the other loses. (There are no ties, and for simplicity, no rain-outs; each game really takes place, and at the end, one team wins and one team loses, without exception.) There is an agreed-upon schedule that specifies which team should play which other team when; in the entire season, each team is scheduled to play, in total, the same number of games. Each team wants to come in first place; that is, it wants to end the schedule having won the most number of games. If the season ends with two or more teams tied for first place, then all of these teams are considered to have come in first place. You follow one of these teams closely and are rooting for them to come in first place; we shall assume that we have indexed the teams so that your team is team n . Unfortunately, it is mid-season and your team is not doing very well at all. You would like to know if it is still possible for them to end the season in first place, or whether, to your great dismay, they have already been eliminated. We shall show that this problem can be formulated as a maximum flow problem. The input for this baseball elimination problem consists of the following. A summary of the remaining games of the season: for each pair of teams i and j , i,j = 1 ,...,n , the number of games remaining to be played between them, which is denoted g ( i,j ) . The standings thus far, which can be summarized by specifying, for each team i = 1 ,...,n , the number of games that team i has won thus far, which shall be denoted w ( i ) . We next wish to make precise what it means for our team n to have been eliminated. We indicate instead what it would mean if team n were not eliminated. How many games would team n win if it goes undefeated for the rest of the season (which is clearly the most rosy scenario)? It currently has w ( n ) wins, and there are n- 1 j =1 g ( j,n ) games remaining on its schedule, and so a perfect completion to its season would give team n with w ( n ) + n- 1 j =1 g ( j,n ) wins in total. Let this number of wins be denoted W . Team n is not eliminated if there is some way that the rest of the games between the other teams can turn out so that each other team has at most W wins at the end of the season. There are g ( i,j ) games remaining between teams i and j . Suppose that x ( i,j ) denotes the number of games between i and j , of those remaining, that team i wins. Note that x ( i,j ) + x ( j,i ) = g ( i,j ) = g ( j,i ) for each i,j = 1 ,... ,n . (Be sure that you understand why this is true; there is little point in reading on before you do! As a further check, what isunderstand why this is true; there is little point in reading on before you do!...
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