Cutting & Packing

Overview

Cutting and packing optimization determines how to cut raw materials or pack items into containers while minimizing waste. Whether you are cutting steel sheets, slicing glass panels, loading pallets into trucks, or arranging products in warehouse bins, the core challenge is the same: fit as many items as possible into limited space with minimal leftover material.

These problems appear across nearly every manufacturing and logistics operation, and even small percentage reductions in waste can save substantial costs at scale.

When to Use This

• Scenario: You have items of varying sizes that must be cut from stock material or packed into fixed-size bins, and you want to minimize the number of bins used or the total wasted material
• Industry: Steel and metal fabrication, glass manufacturing, paper and packaging, warehousing, container shipping
• ROI: Typical waste reduction of 5-20%, translating directly to material cost savings and fewer shipments

Step-by-Step Walkthrough

1. Define item dimensions

List all items to be cut or packed, including their sizes (length, width, or volume depending on the problem). Specify quantities needed for each item type.

2. Specify stock material or container sizes

Define the raw material dimensions (sheet size, roll width) or container capacities (bin volume, truck payload). Note any orientation constraints -- can items be rotated?

3. Set the objective

Choose your goal:

• Minimize bins used -- pack all items into the fewest containers
• Minimize waste -- cut all pieces from the least amount of raw material
• Maximize utilization -- fit as much value as possible into a single container

4. Add problem-specific constraints

Include rules such as:

• No overlapping items
• Weight limits per container
• Fragile items cannot be stacked
• Certain items must stay together or apart

5. Analyze the cutting or packing plan

Review which items go into which bins or cutting patterns. Look for patterns you can reuse across production runs. Identify items that consistently cause waste -- these may benefit from design changes.

Example Parameters

import httpx

API_URL = "https://api.jaot.io/api/v2"
headers = {"Authorization": "Bearer ok_live_your_key_here"}

# Bin packing: assign 6 items to bins of capacity 100
response = httpx.post(f"{API_URL}/solve", headers=headers, json={
    "variables": [
        # x_i_j = 1 if item i is in bin j
        {"name": "item1_bin1", "type": "binary"},
        {"name": "item1_bin2", "type": "binary"},
        {"name": "item2_bin1", "type": "binary"},
        {"name": "item2_bin2", "type": "binary"},
        {"name": "item3_bin1", "type": "binary"},
        {"name": "item3_bin2", "type": "binary"},
        {"name": "item4_bin1", "type": "binary"},
        {"name": "item4_bin2", "type": "binary"},
        {"name": "item5_bin1", "type": "binary"},
        {"name": "item5_bin2", "type": "binary"},
        # y_j = 1 if bin j is used
        {"name": "use_bin1", "type": "binary"},
        {"name": "use_bin2", "type": "binary"},
    ],
    "objective": {
        "sense": "minimize",
        "coefficients": {"use_bin1": 1, "use_bin2": 1},
    },
    "constraints": [
        {
            "name": "item1_assigned",
            "coefficients": {"item1_bin1": 1, "item1_bin2": 1},
            "sense": "==",
            "rhs": 1,
        },
        {
            "name": "item2_assigned",
            "coefficients": {"item2_bin1": 1, "item2_bin2": 1},
            "sense": "==",
            "rhs": 1,
        },
        {
            "name": "bin1_capacity",
            "coefficients": {
                "item1_bin1": 35, "item2_bin1": 25,
                "item3_bin1": 40, "item4_bin1": 20,
                "item5_bin1": 30, "use_bin1": -100
            },
            "sense": "<=",
            "rhs": 0,
        },
    ],
})
result = response.json()

print(f"Bins used: {result['objective_value']:.0f}")

Templates

Next Steps

• Warehouse Layout & Operations -- Optimize storage layouts and picking routes intermediate
• Construction Project Planning -- Apply resource optimization to construction scheduling intermediate
• Textile Manufacturing -- Minimize fabric waste with optimized cutting patterns intermediate