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# ps10 - 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 10, Due May 11 th , 2007 You will need 88 points out of 104 to receive a grade of 5 Problem 1: Help Your Friend Pack! (22 points) At the end of the Spring semester, Doug needs to rent a U-Haul to transport his stuff to a new apartment. In order to do this, he needs to pack some of his items into boxes and then pack everything into the back of a U-Haul truck. Fortunately, he has boxes left over from a previous move. Since Doug is living on a Ph.D. stipend, he’d like fit all of his stuff into as small a truck as possible. The following tables list the different items that need to be packed as well as the available boxes. For this problem, assume that the listed items cannot be split between boxes. Item # Item Size (ft. 3 ) 1 Textbooks 11 2 CDs and DVDs 12 3 Computer 17 4 Board Games 18 5 Electronics 22 6 Sports Equipment 26 Box # Size (ft. 3 ) 1 20 2 20 3 25 4 25 5 30 6 30 Please note that a box does not need to be packed completely full. For instance, a 30 ft. 3 box could be packed with the Computer, CDs and DVDs or with the Board Games, CDs and DVDs. Part A: (4 points) Please formulate an Integer Program to determine if all of the packed items can fit into a truck that provides 120 ft. 3 of space. (Hint: One of the decisions involved is which boxes to use.) Part B: (2 points) Reformulate the Integer Program from Part A to minimize the truck space required to pack all of the boxes. Page 0 of 9

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Part C: (6 points) Solve the Integer Program from Part B using Excel. When optimizing your model in Excel, please configure the following Excel Solver options: Add appropriate ‘binary’ constraints in the ‘Subject to Constraints’ box Select ‘Assume Linear Model’ in the ‘Options’ screen Select ‘Assume Non-Negative’ in the ‘Options’ screen Set ‘Max Time’ to 600 second in the ‘Options’ screen Set ‘Iterations’ to 5000 in the ‘Options’ Screen Set ‘Tolerance’ to 0.01% in the ‘Options’ Screen Remember to add integrality constraints in Excel and lower the solution Part D: (2 points) Based on your Excel solution to Part C , what can you say about the answer to Part A ? Part E: (2 points) The following additional items also need to be put into the moving truck, but do not need to be packed. How would this change your formulation from Part B ? Item # Item Size (ft. 3 ) 7 Desk 45 8 Dressers 50 9 Couch 55 10 Bed 60 Part F: (6 points) For each of the following logical restrictions, please write the constraint or constraints necessary to model the restriction within this problem. (Note: You do not need to rewrite the full Integer Program.) i.) The Textbooks and the Electronics cannot be packed in the same box.
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