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hmk_5_s - ESE304 Introduction to Optimization(Homework#5...

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Unformatted text preview: ESE304 - Introduction to Optimization (Homework #5) Fall Semester, 2010 M. Carchidi –––––––––––––––––––––––––––––––––––– Problem #1 (15 points) Determine the optimal solution to the following MIP problem. max z = 2 x 1 + x 2 (Objective Function) s.t. 7 x 1 + 3 x 2 ≤ 34 (Constraint #1) 14 x 1 + 9 x 2 ≤ 79 (Constraint #2) x 1 = integer (Integer Restrictions) x 1 , x 2 ≥ (Sign Restrictions) You may start with the optimal tableau, Optimal Tableau For The LP Relaxation Problem Row z x 1 x 2 s 1 s 2 rhs BVs [1] 4 / 21 1 / 21 215 / 21 z 1 [1] 3 / 7 − 1 / 7 23 / 7 x 1 2 [1] − 2 / 3 1 / 3 11 / 3 x 2 for the LP relaxation of the above MIP. –––––––––––––––––––––––––––––––––––– –––––––––––––––––––––––––––––––––––– Problem #2 (15 points) A hiker wants to go on a short hike and he must decide what to carry. His choices are to carry: Food, Water, Eating Utensils, a Compass and a Map. The weights of these and their value to the hiker are summarized in the table below. Item # Item Name Value Weight 1 Food 4 4 2 Water 5 4 3 Eating Utensils 1 3 4 Compass 2 2 5 Map 3 1 The hiker can carry no more than 9 units of weight and the hiker will carry either food or water, but not both. In addition, if he carries the compass, then he will want to carry the map, and if he does not carry any food, then he will not carry the eating utensils as well. a.) (10 points) Formulate an IP problem which will maximize the value of the items carried by the hiker. b.) (5 points) Using any method you like (except a method that uses a com- puter), determine a solution to your problem. –––––––––––––––––––––––––––––––––––– 2 –––––––––––––––––––––––––––––––––––– Problem #3 (10 points) Consider the following game. You are to pick out 4 three-letter ”words” from the following list. DBA DEG ADI FFD GHI BCD FDF BAI For each word, you earn a score equal to the position that the word’s third letter appears in the alphabet. For example, the word DBA earns a score of 1 (since A appears f rst in the alphabet), DEG earns you a score of 7 (since G appears seventh in the alphabet), and so on. Your goal is to choose the four words that maximize your total score, subject to the following constraint: The sum of the positions on the alphabet for the f rst letter of each word chosen must be at least as large as the sum of the positions in the alphabet for the second letter of each word chosen. Formulate a 0-1 IP to solve this problem and use LINDO to get the optimal solution....
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This note was uploaded on 03/02/2011 for the course ESE 304 taught by Professor Michaela.carchidi during the Winter '11 term at UPenn.

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hmk_5_s - ESE304 Introduction to Optimization(Homework#5...

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