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There are many rounding methods that round an integer to the nearest integer, all of which are the same except on the half-integers. The sum of integers returned will be close to the sum of the original numbers on average when the distribution of the fraction parts of the numbers are roughly symmetric to $0.5$. One of those methods might be good enough for you in practice.

However, none of them will ensure the rule for sum, "the sum of output integers must be as close to the sum of original numbers as possible". For example, the nearest integer to $0.6$ and $0.7$ is $1$ but the nearest integer to their sum $0.6+0.7=1.3$ is $1$ instead of $1+1 =2$. Hence to satisfy the rule of sum, we cannot guarantee that each integer is mapped to its nearest integer.

Let us pick one of the rounding methods as $\mathcal R$, which will specify the exact meaning of "as close as possible". For methods below, we will ensure that $$\mathcal R(\text{sum of input numbers})=\text{sum of output integers}$$ while trying to round integers to their respective nearest integers fairly.

Offline Method

Suppose numbers $a_1, a_2, \cdots, a_n$ are given.
Let $\{a_i\}=a_i-\lfloor a_i\rfloor$ be the fractional part of $a_i$ and $s = \{a_1\} + \{a_2\} + \cdots + \{a_n\}$.

• Round down $a_i$ to $\lfloor a_i\rfloor$ for $i\le n-\mathcal R(s)$,
• Round up $a_i$ to $\lfloor a_i\rfloor + 1$ otherwise.

To be fair, sort $a_i$ by $\{a_i\}$ before rounding so that, for example, $4.8$ will be rounded up instead of $7.6$ if one of them should be rounded up.

Online Method

Input: a source that produces numbers
Output: numbers rounded to integers
Procedure:

1. Let number $gap$ be $0$
2. For each number $num$ in the source:
   1. Let $num{\_}rounded=\mathcal R(gap + num)$. Output $num{\_}rounded$.
   2. Add $num - num{\_}rounded$ to $gap$.

Buffered Method

We can mix the offline method and the online method by repeating the following procedure after initializing $gap=0$.

1. Apply the offline method to the next block of numbers, including $gap$ as a summand for $s$ as well.
2. Add the difference between the sum of the numbers returned in step 1 and the sum of the original numbers to $gap$.

The size of each block of numbers is up to your choice.