The Asymptotic Worst-Case Ratio of the Bin Packing Problem by Maximum Occupied Space Technique

The bin packing problem (BPP) is an NP-Complete Problem. The problem can be described as there are $N=\{1,2,{\cdots},n\}$ which is a set of item indices and $L=\{s1,s2,{\cdots},sn\}$ be a set of item sizes sj, where $0<sj{\leq}1$, ${\forall}j{\in}N$. The objective is to minimize the number of bins used for packing items in N into a bin such that the total size of items in a bin does not exceed the bin capacity. Assume that the bins have capacity equal to one. In the past, many researchers put on effort to find the heuristic algorithms instead of solving the problem to optimality. Then, the quality of solution may be measured by the asymptotic worst-case ratio or the average-case ratio. The First Fit Decreasing (FFD) is one of the algorithms that its asymptotic worst-case ratio equals to 11/9. Many researchers prove the asymptotic worst-case ratio by using the weighting function and the proof is in a lengthy format. In this study, we found an easier way to prove that the asymptotic worst-case ratio of the First Fit Decreasing (FFD) is not more than 11/9. The proof comes from two ideas which are the occupied space in a bin is more than the size of the item and the occupied space in the optimal solution is less than occupied space in the FFD solution. The occupied space is later called the weighting function. The objective is to determine the maximum occupied space of the heuristics by using integer programming. The maximum value is the key to the asymptotic worst-case ratio.