Distributing objects on containers












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$begingroup$


Suppose we have $n$ containers, each has the ability of holding $f_{i}$ object for $i=1, 2, dots, n$. That means $f_{i}$ is the maximum number of object that the $i$th container can hold.



Now, we have $N$ objects in total. We assume that $displaystyle sum_{i=1}^{n} f_{i} > N$ such that all $N$ objects can actually be filled into the containers.



Question:
How to distribute these objects to the containers such that each container would not reach to its upper limit too quickly?



If $f_{i} equiv f$ for all $i$, then my intuition told me that the $N$ objects must be uniformly distributed to the containers. (seems related to entropy). But what if all $f_{i}$ are different? What is the objective function?






optimization entropy







share|cite|improve this question











$endgroup$

















    0












    $begingroup$


    Suppose we have $n$ containers, each has the ability of holding $f_{i}$ object for $i=1, 2, dots, n$. That means $f_{i}$ is the maximum number of object that the $i$th container can hold.



    Now, we have $N$ objects in total. We assume that $displaystyle sum_{i=1}^{n} f_{i} > N$ such that all $N$ objects can actually be filled into the containers.



    Question:
    How to distribute these objects to the containers such that each container would not reach to its upper limit too quickly?



    If $f_{i} equiv f$ for all $i$, then my intuition told me that the $N$ objects must be uniformly distributed to the containers. (seems related to entropy). But what if all $f_{i}$ are different? What is the objective function?






    optimization entropy







    share|cite|improve this question











    $endgroup$















      0












      0








      0





      $begingroup$


      Suppose we have $n$ containers, each has the ability of holding $f_{i}$ object for $i=1, 2, dots, n$. That means $f_{i}$ is the maximum number of object that the $i$th container can hold.



      Now, we have $N$ objects in total. We assume that $displaystyle sum_{i=1}^{n} f_{i} > N$ such that all $N$ objects can actually be filled into the containers.



      Question:
      How to distribute these objects to the containers such that each container would not reach to its upper limit too quickly?



      If $f_{i} equiv f$ for all $i$, then my intuition told me that the $N$ objects must be uniformly distributed to the containers. (seems related to entropy). But what if all $f_{i}$ are different? What is the objective function?






      optimization entropy







      share|cite|improve this question











      $endgroup$




      Suppose we have $n$ containers, each has the ability of holding $f_{i}$ object for $i=1, 2, dots, n$. That means $f_{i}$ is the maximum number of object that the $i$th container can hold.



      Now, we have $N$ objects in total. We assume that $displaystyle sum_{i=1}^{n} f_{i} > N$ such that all $N$ objects can actually be filled into the containers.



      Question:
      How to distribute these objects to the containers such that each container would not reach to its upper limit too quickly?



      If $f_{i} equiv f$ for all $i$, then my intuition told me that the $N$ objects must be uniformly distributed to the containers. (seems related to entropy). But what if all $f_{i}$ are different? What is the objective function?






      optimization entropy




      optimization entropy






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Jan 9 at 6:39







      K_inverse

















      asked Jan 9 at 6:22









      K_inverseK_inverse

      288213




      288213






















          1 Answer
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          active

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          2












          $begingroup$

          If you want to maximize the time until any bin reaches its maximum limit.



          At any moment of time, when we want to insert an object, look for a bin with at least $2$ slots left.



          That way, if $Nle sum_{i=1}^nf_i-n-1$, the upper limit for every bin is not attained.



          If $N ge sum_{i=1}^nf_i-n$, then we don't have a choice but let one of the bin attain the maximum.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            How did you obtain the inequality constraint?
            $endgroup$
            – K_inverse
            Jan 9 at 7:40










          • $begingroup$
            Intuitively, when we have too many objects, we have no choice but to hit an upper limit. Now, the question is what is having too many objects. My proposed strategy is to avoid hitting any upper limit whenever we can. The first moment when we can't avoid is when every bins has exactly $1$ slot left. Total number of slots is $sum_{i=1}^n f_i$, and the number of bins is $n$.
            $endgroup$
            – Siong Thye Goh
            Jan 9 at 8:02










          • $begingroup$
            Thanks. Apart from just avoiding hitting the upper limit, I would like to distribute the objects such that every containers would have enough buffer. Not sure what are the keys word to search for this problem. My feeling is that it is related to entropy, but I am not sure.
            $endgroup$
            – K_inverse
            Jan 9 at 11:45












          • $begingroup$
            would have enough buffer? is it that some objects must go to certain bins only?
            $endgroup$
            – Siong Thye Goh
            Jan 9 at 12:28










          • $begingroup$
            well yes, there are some constraints on what are the available containers for each object. But I tried to ignore these constraints first for simplicity.
            $endgroup$
            – K_inverse
            Jan 9 at 12:43












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          1 Answer
          1






          active

          oldest

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          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          2












          $begingroup$

          If you want to maximize the time until any bin reaches its maximum limit.



          At any moment of time, when we want to insert an object, look for a bin with at least $2$ slots left.



          That way, if $Nle sum_{i=1}^nf_i-n-1$, the upper limit for every bin is not attained.



          If $N ge sum_{i=1}^nf_i-n$, then we don't have a choice but let one of the bin attain the maximum.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            How did you obtain the inequality constraint?
            $endgroup$
            – K_inverse
            Jan 9 at 7:40










          • $begingroup$
            Intuitively, when we have too many objects, we have no choice but to hit an upper limit. Now, the question is what is having too many objects. My proposed strategy is to avoid hitting any upper limit whenever we can. The first moment when we can't avoid is when every bins has exactly $1$ slot left. Total number of slots is $sum_{i=1}^n f_i$, and the number of bins is $n$.
            $endgroup$
            – Siong Thye Goh
            Jan 9 at 8:02










          • $begingroup$
            Thanks. Apart from just avoiding hitting the upper limit, I would like to distribute the objects such that every containers would have enough buffer. Not sure what are the keys word to search for this problem. My feeling is that it is related to entropy, but I am not sure.
            $endgroup$
            – K_inverse
            Jan 9 at 11:45












          • $begingroup$
            would have enough buffer? is it that some objects must go to certain bins only?
            $endgroup$
            – Siong Thye Goh
            Jan 9 at 12:28










          • $begingroup$
            well yes, there are some constraints on what are the available containers for each object. But I tried to ignore these constraints first for simplicity.
            $endgroup$
            – K_inverse
            Jan 9 at 12:43
















          2












          $begingroup$

          If you want to maximize the time until any bin reaches its maximum limit.



          At any moment of time, when we want to insert an object, look for a bin with at least $2$ slots left.



          That way, if $Nle sum_{i=1}^nf_i-n-1$, the upper limit for every bin is not attained.



          If $N ge sum_{i=1}^nf_i-n$, then we don't have a choice but let one of the bin attain the maximum.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            How did you obtain the inequality constraint?
            $endgroup$
            – K_inverse
            Jan 9 at 7:40










          • $begingroup$
            Intuitively, when we have too many objects, we have no choice but to hit an upper limit. Now, the question is what is having too many objects. My proposed strategy is to avoid hitting any upper limit whenever we can. The first moment when we can't avoid is when every bins has exactly $1$ slot left. Total number of slots is $sum_{i=1}^n f_i$, and the number of bins is $n$.
            $endgroup$
            – Siong Thye Goh
            Jan 9 at 8:02










          • $begingroup$
            Thanks. Apart from just avoiding hitting the upper limit, I would like to distribute the objects such that every containers would have enough buffer. Not sure what are the keys word to search for this problem. My feeling is that it is related to entropy, but I am not sure.
            $endgroup$
            – K_inverse
            Jan 9 at 11:45












          • $begingroup$
            would have enough buffer? is it that some objects must go to certain bins only?
            $endgroup$
            – Siong Thye Goh
            Jan 9 at 12:28










          • $begingroup$
            well yes, there are some constraints on what are the available containers for each object. But I tried to ignore these constraints first for simplicity.
            $endgroup$
            – K_inverse
            Jan 9 at 12:43














          2












          2








          2





          $begingroup$

          If you want to maximize the time until any bin reaches its maximum limit.



          At any moment of time, when we want to insert an object, look for a bin with at least $2$ slots left.



          That way, if $Nle sum_{i=1}^nf_i-n-1$, the upper limit for every bin is not attained.



          If $N ge sum_{i=1}^nf_i-n$, then we don't have a choice but let one of the bin attain the maximum.






          share|cite|improve this answer









          $endgroup$



          If you want to maximize the time until any bin reaches its maximum limit.



          At any moment of time, when we want to insert an object, look for a bin with at least $2$ slots left.



          That way, if $Nle sum_{i=1}^nf_i-n-1$, the upper limit for every bin is not attained.



          If $N ge sum_{i=1}^nf_i-n$, then we don't have a choice but let one of the bin attain the maximum.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Jan 9 at 7:09









          Siong Thye GohSiong Thye Goh

          104k1468120




          104k1468120












          • $begingroup$
            How did you obtain the inequality constraint?
            $endgroup$
            – K_inverse
            Jan 9 at 7:40










          • $begingroup$
            Intuitively, when we have too many objects, we have no choice but to hit an upper limit. Now, the question is what is having too many objects. My proposed strategy is to avoid hitting any upper limit whenever we can. The first moment when we can't avoid is when every bins has exactly $1$ slot left. Total number of slots is $sum_{i=1}^n f_i$, and the number of bins is $n$.
            $endgroup$
            – Siong Thye Goh
            Jan 9 at 8:02










          • $begingroup$
            Thanks. Apart from just avoiding hitting the upper limit, I would like to distribute the objects such that every containers would have enough buffer. Not sure what are the keys word to search for this problem. My feeling is that it is related to entropy, but I am not sure.
            $endgroup$
            – K_inverse
            Jan 9 at 11:45












          • $begingroup$
            would have enough buffer? is it that some objects must go to certain bins only?
            $endgroup$
            – Siong Thye Goh
            Jan 9 at 12:28










          • $begingroup$
            well yes, there are some constraints on what are the available containers for each object. But I tried to ignore these constraints first for simplicity.
            $endgroup$
            – K_inverse
            Jan 9 at 12:43


















          • $begingroup$
            How did you obtain the inequality constraint?
            $endgroup$
            – K_inverse
            Jan 9 at 7:40










          • $begingroup$
            Intuitively, when we have too many objects, we have no choice but to hit an upper limit. Now, the question is what is having too many objects. My proposed strategy is to avoid hitting any upper limit whenever we can. The first moment when we can't avoid is when every bins has exactly $1$ slot left. Total number of slots is $sum_{i=1}^n f_i$, and the number of bins is $n$.
            $endgroup$
            – Siong Thye Goh
            Jan 9 at 8:02










          • $begingroup$
            Thanks. Apart from just avoiding hitting the upper limit, I would like to distribute the objects such that every containers would have enough buffer. Not sure what are the keys word to search for this problem. My feeling is that it is related to entropy, but I am not sure.
            $endgroup$
            – K_inverse
            Jan 9 at 11:45












          • $begingroup$
            would have enough buffer? is it that some objects must go to certain bins only?
            $endgroup$
            – Siong Thye Goh
            Jan 9 at 12:28










          • $begingroup$
            well yes, there are some constraints on what are the available containers for each object. But I tried to ignore these constraints first for simplicity.
            $endgroup$
            – K_inverse
            Jan 9 at 12:43
















          $begingroup$
          How did you obtain the inequality constraint?
          $endgroup$
          – K_inverse
          Jan 9 at 7:40




          $begingroup$
          How did you obtain the inequality constraint?
          $endgroup$
          – K_inverse
          Jan 9 at 7:40












          $begingroup$
          Intuitively, when we have too many objects, we have no choice but to hit an upper limit. Now, the question is what is having too many objects. My proposed strategy is to avoid hitting any upper limit whenever we can. The first moment when we can't avoid is when every bins has exactly $1$ slot left. Total number of slots is $sum_{i=1}^n f_i$, and the number of bins is $n$.
          $endgroup$
          – Siong Thye Goh
          Jan 9 at 8:02




          $begingroup$
          Intuitively, when we have too many objects, we have no choice but to hit an upper limit. Now, the question is what is having too many objects. My proposed strategy is to avoid hitting any upper limit whenever we can. The first moment when we can't avoid is when every bins has exactly $1$ slot left. Total number of slots is $sum_{i=1}^n f_i$, and the number of bins is $n$.
          $endgroup$
          – Siong Thye Goh
          Jan 9 at 8:02












          $begingroup$
          Thanks. Apart from just avoiding hitting the upper limit, I would like to distribute the objects such that every containers would have enough buffer. Not sure what are the keys word to search for this problem. My feeling is that it is related to entropy, but I am not sure.
          $endgroup$
          – K_inverse
          Jan 9 at 11:45






          $begingroup$
          Thanks. Apart from just avoiding hitting the upper limit, I would like to distribute the objects such that every containers would have enough buffer. Not sure what are the keys word to search for this problem. My feeling is that it is related to entropy, but I am not sure.
          $endgroup$
          – K_inverse
          Jan 9 at 11:45














          $begingroup$
          would have enough buffer? is it that some objects must go to certain bins only?
          $endgroup$
          – Siong Thye Goh
          Jan 9 at 12:28




          $begingroup$
          would have enough buffer? is it that some objects must go to certain bins only?
          $endgroup$
          – Siong Thye Goh
          Jan 9 at 12:28












          $begingroup$
          well yes, there are some constraints on what are the available containers for each object. But I tried to ignore these constraints first for simplicity.
          $endgroup$
          – K_inverse
          Jan 9 at 12:43




          $begingroup$
          well yes, there are some constraints on what are the available containers for each object. But I tried to ignore these constraints first for simplicity.
          $endgroup$
          – K_inverse
          Jan 9 at 12:43


















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