So the problem I have is the following : there is a set of N categories of objects, in each category there are M objects, each with a specified value and weight. We have to pick one object from each category so that the weight is <= some given capacity W, and the value is maximum. The task has to be solved using the branch and bounds method. I struggle to understand how is this method supposed to work in this situation. Could you please explain it to me?
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That's a bit too broad. You need to take a stab at it first, and post your code if you're still having trouble. – Tom Karzes Nov 06 '17 at 11:24
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That's the problem - I'm not sure I understand how this method is supposed to work in this situation – Miyavistka Nov 06 '17 at 11:32
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so your question has nothing to do with python in particular... – hoefling Nov 06 '17 at 11:53
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Use the power of google and look at https://en.wikipedia.org/wiki/Branch_and_bound and maybe this gives some insight: http://compalg.inf.elte.hu/~tony/Oktatas/SecondExpert/Chapter24-Branch-6April.pdf – Gijs Den Hollander Nov 06 '17 at 15:04
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This is the multiple choice knapsack problem solved in the 70s by Sinha and Zoltners. [Here](http://pubsonline.informs.org/doi/abs/10.1287/opre.27.3.503?journalCode=opre) is the original reference. If you google around you will find some more stuff. – Ioannis Nov 07 '17 at 11:48
1 Answers
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A short example of what the algorithm should do.
lets say you have 4 items [(weight, value)]= [(3, 5),(8, 10),(1, 2),(4, 5)].
First sort them on there value per weight = [(1, 2),(12, 20),(4, 5),(9, 10)]
and the maximum weight is 16.
starting from the first item make a tree where you either ad or drop a item. At each level in the tree calculate weight, the value and the value which is still left in the three. If the value left + value in a branch is less than maximum value you find, then you close that branch. You also close a branch if the weight is more than aloud.
Below a schematic representation of how it should work.
(value) (0)
(weight) (0)
(value left) (37)
add drop
(1,2) <------- ------>
(2) (0)
(1) (0)
(35) (35)
(20,12) ---------------------------------------------------------------
(22) (2) (20) (0)
(13) (1) (12) (0)
(15) *(15) (15) *(15)
(4,5) -----------------------------------------------------------------------
(27) (22) (25) (20)
(17) (13) (16) (12)
**(10) (10) (10) (10)
(9,10) ---------------------------------------------------------------------------
(31) (20) (35) (25) (30) (20)
(22) (13) (25) (16) (21) (12)
**(0) (0) **(0) (0) **(0) (0)
win
* branch is closed due to that the value+value left< then maximum value in the tree
**
branch is closed due to that the weight is more than the aloud weight.
The benefit of this method is that you reduce computations compared to a brute force method. By starting which items which have the highest value per weight, its very likely that you close of branches quickly and reduce computation time.
hopefully this helps
Gijs Den Hollander
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