Bin packing fundamentals for FBA shipments

If you've ever stared at a pile of unpacked SKUs and a stack of cartons and wondered whether the way you're packing them is actually the cheapest way, you've already encountered the bin-packing problem. It's one of the most heavily-studied problems in computer science, and Amazon's FBA program imposes a particularly nasty variant of it. This guide is the first-principles walkthrough — what the problem is, what makes the FBA variant hard, and how MarginLock's solver approaches it.

The classical problem

The classical bin-packing problem is short to state. You have a set of items of various sizes and a set of bins of fixed capacity. You want to pack every item into a bin without exceeding the bin's capacity, using as few bins as possible.

In one dimension — items have a single "size" attribute and bins have a single "capacity" attribute — the problem is NP-hard. There is no known algorithm that solves it optimally in polynomial time. There are, however, several heuristics that get within a few percent of optimal in practice:

• First-fit: for each item, put it in the first bin that has room. Open a new bin if none does.
• First-fit decreasing: sort the items largest-first, then run first-fit.
• Best-fit decreasing: for each item, put it in the bin that will have the least remaining room after the item goes in.

For most practical inputs, first-fit decreasing is within 11% of optimal and runs in O(n log n).

Why FBA makes it harder

The FBA variant is harder than the textbook problem in three ways.

Multiple capacity dimensions

A box has a weight capacity, a cube capacity, and dimensional weight rules. Amazon penalizes you on whichever dimension you violate first. So your bin capacity isn't a single number — it's a vector, and "fitting" means staying under all three constraints simultaneously.

Mixed-SKU constraints

You can't combine arbitrary SKUs. There are mixed-SKU caps, hazmat restrictions, and per-marketplace rules. The solver has to know which items are compatible before it tries to pack them together.

Dimensional-weight thresholds

Amazon's pricing is non-linear. A box that weighs 49.9 lb is priced differently from a 50.1 lb box. The marginal cost of one extra pound near a threshold is enormous. A naïve "fill the box" heuristic that pushes you across the threshold by half a pound is the single most common bin-packing failure we see in the wild.

How MarginLock packs

We use a modified first-fit-decreasing as the base, then layer two extra rules on top.

1. 1

   Sort items by cube, descending

   Largest items go first. This is the classical heuristic, unchanged.

2. 2

   For each item, find candidate bins

   A bin is a candidate if it can accept the item under all three capacity dimensions (weight, cube, dim-weight) and under the mixed-SKU rules.

3. 3

   Score each candidate against threshold proximity

   Among candidates, prefer the one whose post-pack state stays furthest from a dim-weight or weight pricing threshold. This is the rule that eliminates the "49.9 lb that becomes 50.1 lb" failure mode.

4. 4

   Open a new bin if no candidate qualifies

   Only open a new bin when no existing bin satisfies the threshold-proximity score. This minimizes total bins without crossing thresholds.

Worked example

Consider three SKUs going into a single shipment:

A first-fit-decreasing packer that ignores thresholds would happily put all three into one box at 52 lb. That's a 50-lb overweight violation, and the box will be Amazon-flagged for repacking on intake — which costs you both money and one operational day.

A threshold-aware packer notices that combining all three pushes weight past 50 lb and instead splits into two boxes:

• Box 1: A + B = 30 lb
• Box 2: C = 22 lb

Both boxes are well under the threshold and neither will be reboxed.

What about 3D packing?

We don't currently solve true 3D packing — the volumetric arrangement of items inside a carton — and for FBA in particular this isn't a meaningful gap. Amazon doesn't penalize you for sub-optimal arrangement inside a box, as long as the box itself is under the dim-weight threshold. The 1D bin-packing problem on cube and weight is what actually drives your fee outcomes.

If you sell extreme-shape items (long bats, wide canvases) where the bounding box drives dim-weight aggressively, the solver lets you mark those SKUs and will route them to dedicated cartons.

Further reading

For a deeper look at the bin-packing engine itself, including the metric callouts and the threshold-scoring formula: