knapsack with changing value function
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I recently was looking at the knapsack problem and was wondering, if a slight modification can be done.
Let's say that if my bag is empty, I have values $[x_{10},x_{20},x_{30},... ,x_{n0}]$ for n items. But when item i is chosen as a first in the bag, the values change to $[x_{1i},x_{2i},x_{3i},... ,x_{ni}]$. The values change everytime, when I put new items in the bag. Let's also assume that I have the function, how the values change.
Does anyone know how to modify the algorithm to this case? I saw a version of a "greedy" algorithm and it gave me an idea. I would just divide the value by the weight, choose the item that has the biggest ratio. The values would recalculate again, then I would choose the biggest ratio again etc etc. This however doesn't take into consideration that by choosing a smaller value/weight for item i at time t-1 can yield a bigger value at time t. This might yield a bigger total value/weight then by choosing item j at time t-1 that had a bigger value/weight.
Thanks in advance, hopefully some can help me with this problem!
combinatorics optimization algorithms
asked Jan 18 at 17:34
habbh777habbh777
11
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add a comment |
$begingroup$
I recently was looking at the knapsack problem and was wondering, if a slight modification can be done.
Let's say that if my bag is empty, I have values $[x_{10},x_{20},x_{30},... ,x_{n0}]$ for n items. But when item i is chosen as a first in the bag, the values change to $[x_{1i},x_{2i},x_{3i},... ,x_{ni}]$. The values change everytime, when I put new items in the bag. Let's also assume that I have the function, how the values change.
Does anyone know how to modify the algorithm to this case? I saw a version of a "greedy" algorithm and it gave me an idea. I would just divide the value by the weight, choose the item that has the biggest ratio. The values would recalculate again, then I would choose the biggest ratio again etc etc. This however doesn't take into consideration that by choosing a smaller value/weight for item i at time t-1 can yield a bigger value at time t. This might yield a bigger total value/weight then by choosing item j at time t-1 that had a bigger value/weight.
Thanks in advance, hopefully some can help me with this problem!
combinatorics optimization algorithms
asked Jan 18 at 17:34
habbh777habbh777
11
$endgroup$
$begingroup$
Relevant: stackoverflow.com/questions/38868205/…
$endgroup$
– nathan.j.mcdougall
Jan 18 at 18:10
add a comment |
$begingroup$
I recently was looking at the knapsack problem and was wondering, if a slight modification can be done.
Let's say that if my bag is empty, I have values $[x_{10},x_{20},x_{30},... ,x_{n0}]$ for n items. But when item i is chosen as a first in the bag, the values change to $[x_{1i},x_{2i},x_{3i},... ,x_{ni}]$. The values change everytime, when I put new items in the bag. Let's also assume that I have the function, how the values change.
Does anyone know how to modify the algorithm to this case? I saw a version of a "greedy" algorithm and it gave me an idea. I would just divide the value by the weight, choose the item that has the biggest ratio. The values would recalculate again, then I would choose the biggest ratio again etc etc. This however doesn't take into consideration that by choosing a smaller value/weight for item i at time t-1 can yield a bigger value at time t. This might yield a bigger total value/weight then by choosing item j at time t-1 that had a bigger value/weight.
Thanks in advance, hopefully some can help me with this problem!
combinatorics optimization algorithms
asked Jan 18 at 17:34
habbh777habbh777
11
$endgroup$
I recently was looking at the knapsack problem and was wondering, if a slight modification can be done.
Let's say that if my bag is empty, I have values $[x_{10},x_{20},x_{30},... ,x_{n0}]$ for n items. But when item i is chosen as a first in the bag, the values change to $[x_{1i},x_{2i},x_{3i},... ,x_{ni}]$. The values change everytime, when I put new items in the bag. Let's also assume that I have the function, how the values change.
Does anyone know how to modify the algorithm to this case? I saw a version of a "greedy" algorithm and it gave me an idea. I would just divide the value by the weight, choose the item that has the biggest ratio. The values would recalculate again, then I would choose the biggest ratio again etc etc. This however doesn't take into consideration that by choosing a smaller value/weight for item i at time t-1 can yield a bigger value at time t. This might yield a bigger total value/weight then by choosing item j at time t-1 that had a bigger value/weight.
Thanks in advance, hopefully some can help me with this problem!
combinatorics optimization algorithms
combinatorics optimization algorithms
asked Jan 18 at 17:34
habbh777habbh777
11
asked Jan 18 at 17:34
habbh777habbh777
11
asked Jan 18 at 17:34
habbh777habbh777
11
asked Jan 18 at 17:34
habbh777habbh777
11
asked Jan 18 at 17:34
habbh777habbh777
11
11
$begingroup$
Relevant: stackoverflow.com/questions/38868205/…
$endgroup$
– nathan.j.mcdougall
Jan 18 at 18:10
add a comment |
$begingroup$
Relevant: stackoverflow.com/questions/38868205/…
$endgroup$
– nathan.j.mcdougall
Jan 18 at 18:10
$begingroup$
Relevant: stackoverflow.com/questions/38868205/…
$endgroup$
– nathan.j.mcdougall
Jan 18 at 18:10
$begingroup$
Relevant: stackoverflow.com/questions/38868205/…
$endgroup$
– nathan.j.mcdougall
Jan 18 at 18:10
add a comment |
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$begingroup$
Relevant: stackoverflow.com/questions/38868205/…
$endgroup$
– nathan.j.mcdougall
Jan 18 at 18:10