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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 UHaul 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 UHaul 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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View Full Document 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 NonNegative’ 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.
ii.) The 30 ft.
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This note was uploaded on 01/17/2012 for the course MGMT 15.053 taught by Professor Jamesorli during the Spring '07 term at MIT.
 Spring '07
 JamesOrli
 Management

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