Problem 1 (40 points)
Imagine that you are planning your next vacations. For your trip, first you want to determine which items to pack. To take this decision, you make a list of 30 items from which you want to pick the ones to be packed. For each of these 30 items, you know its weight (in pounds) and its associated intrinsic value (the more you want to take the item, the
higher its “value”), as indicated below (Table 1).
Table 1. Weight and value of each item
ITEM Weight (pounds)
Value
Item 1 4 3
Item 2 5 2
Item 3 5 5
Item 4 9 4
Item 5 9 4
Item 6 4 4
Item 7 7 4
Item 8 3 9
Item 9 6 7
Item 10 5 7
Item 11 7 8
Item 12 8 9
Item 13 3 8
Item 14 8 3
Item 15 4 6
Item 16 3 5
Item 17 7 7
Item 18 9 8
Item 19 6 2
Item 20 7 2
Item 21 6 7
Item 22 8 2
Item 23 7 5
Item 24 9 3
Item 25 3 7
Item 26 4 4
Item 27 10 9
Item 28 4 10
Item 29 3 4
Item 30 8 6
PART I. (20 points)
Assume that for your trip you can only use a single bag. Considering the size of the items, you know you cannot pack more than three items in this bag. Also, due to regulations, maximum weight packed in this bag should be 16 pounds. Considering these constraints, calculate the optimal packing strategy (the one that maximizes the total value of the items packed)
1.1. (2 points) In the optimal solution, is item 8 packed?
A. Yes
B. No
1.2. (2 points) In the optimal solution, is item 10 packed?
A. Yes
B. No
1.3. (2 points) In the optimal solution, is item 12 packed?
A. Yes
B. No
1.4. (2 points) In the optimal solution, is item 18 packed?
A. Yes
B. No
1.5. (2 points) In the optimal solution, is item 26 packed?
A. Yes
B. No
1.6. (5 points) In the optimal packing strategy (the one that maximizes the total value packed) what is the total value associated with the items that you can pack?
A. 15
B. 22
C. 28
D. 35
E. 43
F. 71
1.7. (5 points) In the optimal packing strategy (the one that maximizes the total value packed) what is the total weight associated with the items that you can pack in your bag?
A. 11
B. 12
C. 13
D. 14
E. 15
F. 16
PART II. (20 points)
Now, assume that for your trip you can use two bags, where you cannot pack more than three items in the first bag, or more than six in the second bag. Also, due to regulations, maximum weight packed are 16 pounds and 29 pounds for the first bag and second bag, respectively. Considering these constraints, calculate the optimal packing strategy (the one that maximizes the total value of the items packed).
1.8. (2 points) In the new optimal solution, is item 8 packed?
A. Yes
B. No
1.9. (2 points) In the new optimal solution, is item 10 packed?
A. Yes
B. No
1.10. (2 points) In the new optimal solution, is item 12 packed?
A. Yes
B. No
1.11. (2 points) In the new optimal solution, is item 18 packed?
A. Yes
B. No
1.12. (2 points) In the new optimal solution, is item 26 packed?
A. Yes
B. No
1.13. (5 points) In the optimal packing strategy (the one that maximizes the total value packed) what is the total value associated with the items that you can pack?
A. 69
B. 72
C. 74
D. 79
E. 82
F. 85
1.14. (5 points) In the optimal packing strategy (the one that maximizes the total value packed) what is the total weight associated with the items that you can pack in the first bag?
A. 11
B. 12
C. 13
D. 14
E. 15
F. 16
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