Bin Packing Calculator
Solve 1D Bin Packing Problems using First Fit Decreasing (FFD) and Best Fit Decreasing (BFD) heuristics with theoretical minimum lower bounds.
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What Is a Bin Packing Calculator?
The Bin Packing Calculator solves the classic one-dimensional Bin Packing Problem, an NP-hard combinatorial optimization problem where items of different sizes must be packed into a minimum number of fixed-capacity bins.
How to Use This Calculator
- Enter the uniform bin capacity (C).
- Enter a comma-separated list of item sizes/weights.
- Review the solution generated by First Fit Decreasing (FFD) and Best Fit Decreasing (BFD) heuristics.
- Compare the heuristic bin count against the theoretical lower bound ⌈Σw / C⌉.
Theoretical Lower Bound & FFD Guarantee
B_{\text{min}} \ge \left\lceil \frac{\sum_{i=1}^n w_i}{C} \right\rceil, \quad \text{FFD}(I) \le \frac{11}{9} \text{OPT}(I) + \frac{6}{9}Where w_i are item weights and C is bin capacity. The First Fit Decreasing heuristic is guaranteed never to use more than 11/9 of the optimal bin count.
Worked Example
Scenario: Packing items [5, 4, 3, 2, 7, 8, 1, 6, 2, 4] into bins of capacity 10
Total item weight: 5 + 4 + 3 + 2 + 7 + 8 + 1 + 6 + 2 + 4 = 42.
Theoretical minimum bins: ⌈42 / 10⌉ = 5 bins.
Sort descending: [8, 7, 6, 5, 4, 4, 3, 2, 2, 1].
FFD assignment: Bin 1: [8, 2], Bin 2: [7, 3], Bin 3: [6, 4], Bin 4: [5, 4, 1], Bin 5: [2].
Tips & Key Notes
- First Fit Decreasing (FFD) almost always yields the theoretical minimum or at most 1 additional bin in real-world applications.
- Use this algorithm for cut-length optimization (e.g. cutting 10-foot lumber or metal pipes into specified pieces with minimal scrap).
- Can be applied to cloud server virtual machine placement to minimize active server instances.
Frequently Asked Questions
What is the Bin Packing Problem?
It is a foundational computer science optimization problem: given items of varying sizes, pack them into the minimum number of identical bins of fixed capacity.
Why is First Fit Decreasing (FFD) preferred?
Sorting items from largest to smallest before packing prevents large items from being stranded late in the process, delivering solutions within ~22% of optimal in polynomial time.
Where is bin packing used in industry?
Common applications include logistics cargo loading, cloud computing server virtualization (packing virtual machines onto physical servers), and cutting stock problems in paper/steel mills.
What is the difference between First Fit and Best Fit?
First Fit places an item in the first bin that can hold it. Best Fit searches for the bin with the tightest remaining space that can accommodate the item.
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