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# ps9 - MASSACHUSETTS INSTITUTE OF TECHNOLOGY 15.053...

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M ASSACHUSETTS I NSTITUTE OF T ECHNOLOGY 15.053 –Optimization Methods in Management Science (Spring 2007) Problem Set 9, Due May 3 rd , 2007 You will need 84 points out of 161 to receive a grade of 5. Problem 1: Dream Team: IP Formulation (20 Points) Coach Bob is faced with the decision of selecting 7 star players for the “Dream Team”. He has narrowed his choice down to 10 players. For each player, Bob has collected and rated some statistics (1 being best, and 5 being worst) for the players. In addition, players can only play certain positions of the line up. The positions that each player is allowed to play and the player’s assists, scoring, defense and rebound skills are listed in the table below. In order to have a well-rounded team, the coach knows he must fulfill the following requirements: 1. At least two members must be able to play guard, at least four members must be able to play forward, and at least two players must be able to play center (some players have to be versatile). 2. The average assists, scoring, and rebounding level of the 7 star players must be at least 4. ( Keep in mind, 1 is best and 5 is worst) 3. If player 4 and player 6 both play, then player 5 can not be on the team (Players have compatibility issues!). 4. Players 3 and 9 must be selected together because they feel they are most effective when they play together (so either both or neither are selected). 5. Either player 4 or player 3 (or both) must be included because they are the ones that bring in the fans. Given these constraints, Coach Bob wants to maximize the total scoring ability of the “Dream team”. Formulate an IP that will help him choose his starting team. (Do not solve the IP) Player Position Assists Scoring Rebounding Defense 1 G 3 4 2 1 2 C 2 1 3 4 3 G-F 4 2 2 4 4 F-C 1 3 3 1 5 G-F 5 2 1 2 6 F-C 4 1 2 3 7 G-F 3 5 3 1 8 G-C 2 3 4 1 9 F 2 2 2 5 10 G-F 3 3 1 2 Problem 2: IP Constraints (30 Points; 5 Points Per Part) Page 0 of 6

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Part A: How can integer programming be used to ensure that the variable x can assume only the values 3, 5, 7, and 9?
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ps9 - MASSACHUSETTS INSTITUTE OF TECHNOLOGY 15.053...

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