Bin Packing Algorithms

• Capacity conservation principle: The total used capacity across all bins must at least equal the total object size, which gives a universal lower limit on bin count. This is why a simple arithmetic bound is powerful before any packing starts. It provides a benchmark that no algorithm can beat.

• Lower-bound formula: The standard bound is

Lower bound: $LB = \left\lceil \frac{\sum_{i=1}^{n} s_i}{C} \right\rceil$

where $s_i$ is object size, $n$ is number of objects, and $C$ is bin capacity. The ceiling is essential because a fractional bin is impossible, so any non-integer quotient must round up. If an algorithm uses exactly $LB$ bins, the solution is proven optimal.

• Order sensitivity: Greedy placement decisions depend on object order, so two orderings of the same objects can produce different numbers of bins. This explains why first-fit can be fast but inconsistent in quality across datasets. Sorting before packing often improves structure and reduces wasted residual spaces.

First-Fit (FF)

• Procedure: Process objects in given order and place each object into the first existing bin with enough remaining capacity; open a new bin only if none can accept it. This rule is fast because it avoids pre-sorting and scans bins in a fixed order. It is best when throughput is critical and near-optimal packing is acceptable.

First-Fit Decreasing (FFD)

• Procedure: Sort objects in descending size first, then apply first-fit exactly as above. Placing large items early reduces fragmentation, because early bins reserve less awkward leftover capacity. This usually improves bin utilization compared with FF at the cost of sorting overhead.

Full-Bin + First-Fit Hybrid

• Procedure: First identify combinations that exactly fill capacity $C$, then apply first-fit to remaining objects. Exact fills remove waste immediately and can push the result closer to the lower bound. This method is often stronger in solution quality but more inspection-heavy and slower for large lists.

• Algorithm comparison: Choosing a bin packing heuristic is mainly a trade-off between speed, preprocessing effort, and closeness to optimal. Faster methods accept more order sensitivity, while stronger methods spend effort to reduce fragmentation. The table gives a decision view you can apply quickly.

• Lower bound vs achieved bins: The lower bound is a theoretical minimum threshold, while the achieved bin count is the algorithm's actual output. If achieved bins exceed $LB$, the solution may still be good but is not provably optimal. If achieved bins equal $LB$, you can conclude optimality immediately.

• Exact-fill search vs greedy placement: Full-bin logic targets global structure by consuming combinations that perfectly hit capacity, while first-fit variants make local greedy choices. Global structure often improves final quality because it prevents leftover gaps that are too small to use later. Greedy placement is still valuable because it scales well and is simple to execute accurately.

• Use a strict routine: Start by computing $LB$, then apply the specified algorithm exactly as named, and finally state the number of bins clearly. This sequence earns method marks because it shows both theory and execution. It also gives a built-in reasonableness check when you compare output to $LB$.

• Track bin state explicitly: Record either used capacity or remaining capacity beside each bin after each placement. This prevents arithmetic slips and makes first-available checks transparent. It also helps you avoid illegal placements that exceed capacity by a small amount.

• Finish with a completion statement: Explicitly confirm all objects are assigned and no bin violates capacity. Then state whether the result is optimal by testing if used bins equal the lower bound. Examiners reward this final logical closure because it proves understanding, not just arithmetic.

• Rounding the lower bound incorrectly: A common error is rounding $\frac{\sum s_i}{C}$ to the nearest integer instead of rounding up. That understates the minimum bins and leads to false optimality claims. Always apply the ceiling function because bin counts are discrete.

• Confusing first-fit with best-fit behavior: First-fit chooses the first feasible bin in order, not the tightest-fitting bin by leftover space. Re-optimizing each step by intuition changes the algorithm and can lose method credit in assessments. Follow the bin order rule exactly to preserve correctness of the named method.

• Ignoring completion conditions in full-bin packing: Some learners stop after finding several full combinations and forget to place leftovers with first-fit. That leaves the solution incomplete even if early bins look efficient. The method is only complete when every object appears in exactly one bin.