Bin Packing & Knapsack Solver

Problem variants

1D Bin Packing

Pack items of varying weights into bins of fixed capacity. Minimize number of bins.

Knapsack

Select items with values and weights to maximize total value within a weight budget.

2D Bin Packing

Pack rectangles into sheets. Common in cutting stock and container loading.

Set Cover / Partition

Cover a universe of elements with minimum subsets from a collection.

Bin packing example

python

import nerox
import numpy as np

client = nerox.Client()

bin_capacity = 100
item_weights = [34, 23, 67, 12, 45, 78, 9, 55, 33, 21, 60, 48]

job = client.optimize.bin_packing(
    item_weights=item_weights,
    bin_capacity=bin_capacity,
    solver="gpu",
)

result = job.wait()
print(f"Bins used: {result.objective:.0f}")  # minimize this
for bin_idx, items in enumerate(result.bins):
    total = sum(item_weights[i] for i in items)
    print(f"Bin {bin_idx}: items={items}  load={total}/{bin_capacity}")

Knapsack example

python

# 0/1 Knapsack: maximize value subject to weight capacity
values  = [10, 6, 8, 14, 5, 9, 12, 3, 7, 15]
weights = [5,  4, 6,  8, 3, 7,  9, 2, 5, 10]
capacity = 25

job = client.optimize.knapsack(
    values=values,
    weights=weights,
    capacity=capacity,
    solver="gpu",
)

result = job.wait()
selected = [i for i, x in enumerate(result.solution) if x]
print(f"Selected items: {selected}")
print(f"Total value: {sum(values[i] for i in selected)}")
print(f"Total weight: {sum(weights[i] for i in selected)}")

Applications

Cargo loading — pack containers to minimize shipping trips

Cloud resource allocation — assign workloads to servers to minimize instance count

Cutting stock — minimize material waste in sheet/roll cutting

Financial lot sizing — round fractional positions to integer share quantities

Network traffic engineering — assign flows to links within bandwidth limits