Partitioning to solve Bin Packing Problems

Abstract

The Bin Packing Problem involves efficiently packing items into a limited number of bins without exceeding their capacity. In this paper, we try to answer a specific question in this field. Mathematically the combinatorial optimization problem of this classical Bin Packing can be formulated as follows "Will it be possible to distribute $n$ items with sizes $a_1, \cdots ,a_n$ in total $k$ numbers of bins such that each of the bins consists of exactly $\ell$ number of items whose combined item size is $\mathbb{C}$ provided none of the items left after the allocation? In other words, can we achieve the optimal arrangement $\mathbb{C} =\sum_{i=1}^{n} a_i$ such that $k\ell=n$ also holds under the given size restriction on bins?" We propose an algorithm that determines whether a Bin Packing arrangement is possible while ensuring that each packed bin is distinct. We use tools from standard mathematical concepts such as partitioning of natural number and multiset to derive the algorithm. This algorithm is efficient, easy to implement, and ensures the desired packing in all applicable cases.