FA-95156 / Warehouse slotting and bin packing / Open access

Nested bin fill: empty lines consume volume · case 01

A zero-quantity order line adds negative or phantom volume and uses a SKU slot.

Verified by executionVariant 1 · 8 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

Zero-quantity SKUs are not skipped before volume and SKU count are accumulated.

VERIFIED REPAIR

Skip SKUs whose merged quantity is zero before counting them.

Unsuccessful approach: Skipping only the volume still counts the empty SKU against the SKU limit.

Case contract

lines [[sku, qty, unit_volume, nest_increment]]. Lines for the same SKU are merged. A nesting SKU (increment > 0) occupies unit_volume + (qty-1)*increment; others qty*unit_volume; qty 0 contributes nothing and is not counted. Bin fits when total <= bin_vol and distinct stocked SKUs <= max_skus. Return [total, fits].

Why this case matters

Nesting products such as cups and trays occupy far less space than unit cube suggests.

1 / The failure

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json

N = 1
observations = []
def solve(lines, bin_vol, max_skus):
    merged = {}
    for sku, qty, uv, nv in lines:
        if sku in merged:
            merged[sku][0] += qty
        else:
            merged[sku] = [qty, uv, nv]
    total = 0
    count = 0
    for sku, (qty, uv, nv) in merged.items():
        count += 1
        if nv > 0:
            total += uv + (qty - 1) * nv
        else:
            total += qty * uv
    return [total, total <= bin_vol and count <= max_skus]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['zero line', [[['CUP', 1, 10, 2], ['BOX', 0, 40, 0]], 30, 1], [10, True]], ['generated mix 5', [[['TRY', 10, 25, 5], ['LID', 0, 4, 0]], 120, 1], [70, True]], ['nesting', [[['CUP', 10, 10, 2]], 30, 1], [28, True]], ['merged lines nest together', [[['CUP', 5, 10, 2], ['CUP', 5, 10, 2]], 30, 1], [28, True]], ['sku limit', [[['CUP', 1, 10, 2], ['LID', 1, 4, 0]], 100, 1], [14, False]], ['capacity equal', [[['CUP', 11, 10, 2]], 30, 1], [30, True]], ['non nesting', [[['BOX', 2, 40, 0]], 80, 1], [80, True]], ['generated mix 9', [[['CUP', 10, 10, 2], ['TRY', 10, 25, 5], ['BOX', 5, 40, 0], ['LID', 10, 4, 0], ['BWL', 2, 30, 6]], 80, 1], [374, False]]], [['generated mix 28', [[['TRY', 0, 25, 5]], 120, 1], [0, True]], ['generated mix 5', [[['TRY', 10, 25, 5], ['LID', 0, 4, 0]], 120, 1], [70, True]], ['zero line', [[['CUP', 1, 10, 2], ['BOX', 0, 40, 0]], 30, 1], [10, True]], ['sku limit', [[['CUP', 1, 10, 2], ['LID', 1, 4, 0]], 100, 1], [14, False]], ['capacity equal', [[['CUP', 11, 10, 2]], 30, 1], [30, True]], ['generated mix 10', [[['CUP', 10, 10, 2], ['BOX', 0, 40, 0], ['BOX', 1, 40, 0]], 80, 1], [68, False]], ['generated mix 18', [[['CUP', 10, 10, 2]], 150, 2], [28, True]], ['generated mix 26', [[['CUP', 5, 10, 2], ['BOX', 1, 40, 0], ['BOX', 2, 40, 0]], 50, 3], [138, False]]], [['generated mix 57', [[['BWL', 0, 30, 6]], 120, 3], [0, True]], ['generated mix 5', [[['TRY', 10, 25, 5], ['LID', 0, 4, 0]], 120, 1], [70, True]], ['nesting', [[['CUP', 10, 10, 2]], 30, 1], [28, True]], ['merged lines nest together', [[['CUP', 5, 10, 2], ['CUP', 5, 10, 2]], 30, 1], [28, True]], ['non nesting', [[['BOX', 2, 40, 0]], 80, 1], [80, True]], ['generated mix 21', [[['BWL', 1, 30, 6], ['CUP', 5, 10, 2], ['BOX', 0, 40, 0], ['TRY', 5, 25, 5], ['BWL', 2, 30, 6]], 100, 2], [105, False]], ['generated mix 29', [[['TRY', 0, 25, 5], ['LID', 2, 4, 0], ['BOX', 5, 40, 0]], 150, 2], [208, False]], ['generated mix 37', [[['BWL', 5, 30, 6], ['BOX', 5, 40, 0], ['LID', 1, 4, 0], ['LID', 2, 4, 0]], 100, 2], [266, False]]], [['generated mix 19', [[['BOX', 2, 40, 0], ['TRY', 0, 25, 5]], 80, 3], [80, True]], ['generated mix 5', [[['TRY', 10, 25, 5], ['LID', 0, 4, 0]], 120, 1], [70, True]], ['zero line', [[['CUP', 1, 10, 2], ['BOX', 0, 40, 0]], 30, 1], [10, True]], ['sku limit', [[['CUP', 1, 10, 2], ['LID', 1, 4, 0]], 100, 1], [14, False]], ['capacity equal', [[['CUP', 11, 10, 2]], 30, 1], [30, True]], ['generated mix 33', [[['LID', 5, 4, 0], ['CUP', 1, 10, 2], ['TRY', 0, 25, 5], ['BWL', 10, 30, 6]], 50, 3], [114, False]], ['generated mix 41', [[['BOX', 2, 40, 0]], 120, 2], [80, True]], ['generated mix 49', [[['BWL', 0, 30, 6], ['LID', 1, 4, 0], ['LID', 10, 4, 0], ['BWL', 10, 30, 6]], 120, 1], [128, False]]], [['generated mix 51', [[['LID', 10, 4, 0], ['TRY', 0, 25, 5], ['BWL', 10, 30, 6]], 150, 2], [124, True]], ['generated mix 5', [[['TRY', 10, 25, 5], ['LID', 0, 4, 0]], 120, 1], [70, True]], ['nesting', [[['CUP', 10, 10, 2]], 30, 1], [28, True]], ['merged lines nest together', [[['CUP', 5, 10, 2], ['CUP', 5, 10, 2]], 30, 1], [28, True]], ['non nesting', [[['BOX', 2, 40, 0]], 80, 1], [80, True]], ['generated mix 43', [[['LID', 2, 4, 0], ['TRY', 5, 25, 5]], 100, 2], [53, True]], ['generated mix 52', [[['CUP', 1, 10, 2]], 120, 3], [10, True]], ['generated mix 60', [[['LID', 1, 4, 0], ['BWL', 10, 30, 6]], 120, 2], [88, True]]]]
for label, args, expected in fixtures[N-1]:
    check(label, solve(*args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)

Boundary fixture	Actual	Expected	Outcome
zero line	`[10, False]`	`[10, True]`	Failed
generated mix 5	`[70, False]`	`[70, True]`	Failed
nesting	`[28, True]`	`[28, True]`	Passed
merged lines nest together	`[28, True]`	`[28, True]`	Passed
sku limit	`[14, False]`	`[14, False]`	Passed
capacity equal	`[30, True]`	`[30, True]`	Passed
non nesting	`[80, True]`	`[80, True]`	Passed
generated mix 9	`[374, False]`	`[374, False]`	Passed

SHA-256 / e38274241999214b8367462a09ae4ffe1069e7b828347ca47d9b9f8d7ff8da0e

2 / The unsuccessful fix

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json

N = 1
observations = []
def solve(lines, bin_vol, max_skus):
    merged = {}
    for sku, qty, uv, nv in lines:
        if sku in merged:
            merged[sku][0] += qty
        else:
            merged[sku] = [qty, uv, nv]
    total = 0
    count = 0
    for sku, (qty, uv, nv) in merged.items():
        count += 1
        if qty <= 0:
            continue
        if nv > 0:
            total += uv + (qty - 1) * nv
        else:
            total += qty * uv
    return [total, total <= bin_vol and count <= max_skus]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['zero line', [[['CUP', 1, 10, 2], ['BOX', 0, 40, 0]], 30, 1], [10, True]], ['generated mix 5', [[['TRY', 10, 25, 5], ['LID', 0, 4, 0]], 120, 1], [70, True]], ['nesting', [[['CUP', 10, 10, 2]], 30, 1], [28, True]], ['merged lines nest together', [[['CUP', 5, 10, 2], ['CUP', 5, 10, 2]], 30, 1], [28, True]], ['sku limit', [[['CUP', 1, 10, 2], ['LID', 1, 4, 0]], 100, 1], [14, False]], ['capacity equal', [[['CUP', 11, 10, 2]], 30, 1], [30, True]], ['non nesting', [[['BOX', 2, 40, 0]], 80, 1], [80, True]], ['generated mix 9', [[['CUP', 10, 10, 2], ['TRY', 10, 25, 5], ['BOX', 5, 40, 0], ['LID', 10, 4, 0], ['BWL', 2, 30, 6]], 80, 1], [374, False]]], [['generated mix 28', [[['TRY', 0, 25, 5]], 120, 1], [0, True]], ['generated mix 5', [[['TRY', 10, 25, 5], ['LID', 0, 4, 0]], 120, 1], [70, True]], ['zero line', [[['CUP', 1, 10, 2], ['BOX', 0, 40, 0]], 30, 1], [10, True]], ['sku limit', [[['CUP', 1, 10, 2], ['LID', 1, 4, 0]], 100, 1], [14, False]], ['capacity equal', [[['CUP', 11, 10, 2]], 30, 1], [30, True]], ['generated mix 10', [[['CUP', 10, 10, 2], ['BOX', 0, 40, 0], ['BOX', 1, 40, 0]], 80, 1], [68, False]], ['generated mix 18', [[['CUP', 10, 10, 2]], 150, 2], [28, True]], ['generated mix 26', [[['CUP', 5, 10, 2], ['BOX', 1, 40, 0], ['BOX', 2, 40, 0]], 50, 3], [138, False]]], [['generated mix 57', [[['BWL', 0, 30, 6]], 120, 3], [0, True]], ['generated mix 5', [[['TRY', 10, 25, 5], ['LID', 0, 4, 0]], 120, 1], [70, True]], ['nesting', [[['CUP', 10, 10, 2]], 30, 1], [28, True]], ['merged lines nest together', [[['CUP', 5, 10, 2], ['CUP', 5, 10, 2]], 30, 1], [28, True]], ['non nesting', [[['BOX', 2, 40, 0]], 80, 1], [80, True]], ['generated mix 21', [[['BWL', 1, 30, 6], ['CUP', 5, 10, 2], ['BOX', 0, 40, 0], ['TRY', 5, 25, 5], ['BWL', 2, 30, 6]], 100, 2], [105, False]], ['generated mix 29', [[['TRY', 0, 25, 5], ['LID', 2, 4, 0], ['BOX', 5, 40, 0]], 150, 2], [208, False]], ['generated mix 37', [[['BWL', 5, 30, 6], ['BOX', 5, 40, 0], ['LID', 1, 4, 0], ['LID', 2, 4, 0]], 100, 2], [266, False]]], [['generated mix 19', [[['BOX', 2, 40, 0], ['TRY', 0, 25, 5]], 80, 3], [80, True]], ['generated mix 5', [[['TRY', 10, 25, 5], ['LID', 0, 4, 0]], 120, 1], [70, True]], ['zero line', [[['CUP', 1, 10, 2], ['BOX', 0, 40, 0]], 30, 1], [10, True]], ['sku limit', [[['CUP', 1, 10, 2], ['LID', 1, 4, 0]], 100, 1], [14, False]], ['capacity equal', [[['CUP', 11, 10, 2]], 30, 1], [30, True]], ['generated mix 33', [[['LID', 5, 4, 0], ['CUP', 1, 10, 2], ['TRY', 0, 25, 5], ['BWL', 10, 30, 6]], 50, 3], [114, False]], ['generated mix 41', [[['BOX', 2, 40, 0]], 120, 2], [80, True]], ['generated mix 49', [[['BWL', 0, 30, 6], ['LID', 1, 4, 0], ['LID', 10, 4, 0], ['BWL', 10, 30, 6]], 120, 1], [128, False]]], [['generated mix 51', [[['LID', 10, 4, 0], ['TRY', 0, 25, 5], ['BWL', 10, 30, 6]], 150, 2], [124, True]], ['generated mix 5', [[['TRY', 10, 25, 5], ['LID', 0, 4, 0]], 120, 1], [70, True]], ['nesting', [[['CUP', 10, 10, 2]], 30, 1], [28, True]], ['merged lines nest together', [[['CUP', 5, 10, 2], ['CUP', 5, 10, 2]], 30, 1], [28, True]], ['non nesting', [[['BOX', 2, 40, 0]], 80, 1], [80, True]], ['generated mix 43', [[['LID', 2, 4, 0], ['TRY', 5, 25, 5]], 100, 2], [53, True]], ['generated mix 52', [[['CUP', 1, 10, 2]], 120, 3], [10, True]], ['generated mix 60', [[['LID', 1, 4, 0], ['BWL', 10, 30, 6]], 120, 2], [88, True]]]]
for label, args, expected in fixtures[N-1]:
    check(label, solve(*args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)

Boundary fixture	Actual	Expected	Outcome
zero line	`[10, False]`	`[10, True]`	Failed
generated mix 5	`[70, False]`	`[70, True]`	Failed
nesting	`[28, True]`	`[28, True]`	Passed
merged lines nest together	`[28, True]`	`[28, True]`	Passed
sku limit	`[14, False]`	`[14, False]`	Passed
capacity equal	`[30, True]`	`[30, True]`	Passed
non nesting	`[80, True]`	`[80, True]`	Passed
generated mix 9	`[374, False]`	`[374, False]`	Passed

SHA-256 / 1309da36c4dae33b81b03d933b16f0fa51c760436085c8f6e9a95a8a8d897a80

3 / The verified repair

Exit 0

"""Failure Map reference implementation. Python standard library only."""
import json

N = 1
observations = []
def solve(lines, bin_vol, max_skus):
    merged = {}
    for sku, qty, uv, nv in lines:
        if sku in merged:
            merged[sku][0] += qty
        else:
            merged[sku] = [qty, uv, nv]
    total = 0
    count = 0
    for sku, (qty, uv, nv) in merged.items():
        if qty <= 0:
            continue
        count += 1
        if nv > 0:
            total += uv + (qty - 1) * nv
        else:
            total += qty * uv
    return [total, total <= bin_vol and count <= max_skus]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['zero line', [[['CUP', 1, 10, 2], ['BOX', 0, 40, 0]], 30, 1], [10, True]], ['generated mix 5', [[['TRY', 10, 25, 5], ['LID', 0, 4, 0]], 120, 1], [70, True]], ['nesting', [[['CUP', 10, 10, 2]], 30, 1], [28, True]], ['merged lines nest together', [[['CUP', 5, 10, 2], ['CUP', 5, 10, 2]], 30, 1], [28, True]], ['sku limit', [[['CUP', 1, 10, 2], ['LID', 1, 4, 0]], 100, 1], [14, False]], ['capacity equal', [[['CUP', 11, 10, 2]], 30, 1], [30, True]], ['non nesting', [[['BOX', 2, 40, 0]], 80, 1], [80, True]], ['generated mix 9', [[['CUP', 10, 10, 2], ['TRY', 10, 25, 5], ['BOX', 5, 40, 0], ['LID', 10, 4, 0], ['BWL', 2, 30, 6]], 80, 1], [374, False]]], [['generated mix 28', [[['TRY', 0, 25, 5]], 120, 1], [0, True]], ['generated mix 5', [[['TRY', 10, 25, 5], ['LID', 0, 4, 0]], 120, 1], [70, True]], ['zero line', [[['CUP', 1, 10, 2], ['BOX', 0, 40, 0]], 30, 1], [10, True]], ['sku limit', [[['CUP', 1, 10, 2], ['LID', 1, 4, 0]], 100, 1], [14, False]], ['capacity equal', [[['CUP', 11, 10, 2]], 30, 1], [30, True]], ['generated mix 10', [[['CUP', 10, 10, 2], ['BOX', 0, 40, 0], ['BOX', 1, 40, 0]], 80, 1], [68, False]], ['generated mix 18', [[['CUP', 10, 10, 2]], 150, 2], [28, True]], ['generated mix 26', [[['CUP', 5, 10, 2], ['BOX', 1, 40, 0], ['BOX', 2, 40, 0]], 50, 3], [138, False]]], [['generated mix 57', [[['BWL', 0, 30, 6]], 120, 3], [0, True]], ['generated mix 5', [[['TRY', 10, 25, 5], ['LID', 0, 4, 0]], 120, 1], [70, True]], ['nesting', [[['CUP', 10, 10, 2]], 30, 1], [28, True]], ['merged lines nest together', [[['CUP', 5, 10, 2], ['CUP', 5, 10, 2]], 30, 1], [28, True]], ['non nesting', [[['BOX', 2, 40, 0]], 80, 1], [80, True]], ['generated mix 21', [[['BWL', 1, 30, 6], ['CUP', 5, 10, 2], ['BOX', 0, 40, 0], ['TRY', 5, 25, 5], ['BWL', 2, 30, 6]], 100, 2], [105, False]], ['generated mix 29', [[['TRY', 0, 25, 5], ['LID', 2, 4, 0], ['BOX', 5, 40, 0]], 150, 2], [208, False]], ['generated mix 37', [[['BWL', 5, 30, 6], ['BOX', 5, 40, 0], ['LID', 1, 4, 0], ['LID', 2, 4, 0]], 100, 2], [266, False]]], [['generated mix 19', [[['BOX', 2, 40, 0], ['TRY', 0, 25, 5]], 80, 3], [80, True]], ['generated mix 5', [[['TRY', 10, 25, 5], ['LID', 0, 4, 0]], 120, 1], [70, True]], ['zero line', [[['CUP', 1, 10, 2], ['BOX', 0, 40, 0]], 30, 1], [10, True]], ['sku limit', [[['CUP', 1, 10, 2], ['LID', 1, 4, 0]], 100, 1], [14, False]], ['capacity equal', [[['CUP', 11, 10, 2]], 30, 1], [30, True]], ['generated mix 33', [[['LID', 5, 4, 0], ['CUP', 1, 10, 2], ['TRY', 0, 25, 5], ['BWL', 10, 30, 6]], 50, 3], [114, False]], ['generated mix 41', [[['BOX', 2, 40, 0]], 120, 2], [80, True]], ['generated mix 49', [[['BWL', 0, 30, 6], ['LID', 1, 4, 0], ['LID', 10, 4, 0], ['BWL', 10, 30, 6]], 120, 1], [128, False]]], [['generated mix 51', [[['LID', 10, 4, 0], ['TRY', 0, 25, 5], ['BWL', 10, 30, 6]], 150, 2], [124, True]], ['generated mix 5', [[['TRY', 10, 25, 5], ['LID', 0, 4, 0]], 120, 1], [70, True]], ['nesting', [[['CUP', 10, 10, 2]], 30, 1], [28, True]], ['merged lines nest together', [[['CUP', 5, 10, 2], ['CUP', 5, 10, 2]], 30, 1], [28, True]], ['non nesting', [[['BOX', 2, 40, 0]], 80, 1], [80, True]], ['generated mix 43', [[['LID', 2, 4, 0], ['TRY', 5, 25, 5]], 100, 2], [53, True]], ['generated mix 52', [[['CUP', 1, 10, 2]], 120, 3], [10, True]], ['generated mix 60', [[['LID', 1, 4, 0], ['BWL', 10, 30, 6]], 120, 2], [88, True]]]]
for label, args, expected in fixtures[N-1]:
    check(label, solve(*args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)

Boundary fixture	Actual	Expected	Outcome
zero line	`[10, True]`	`[10, True]`	Passed
generated mix 5	`[70, True]`	`[70, True]`	Passed
nesting	`[28, True]`	`[28, True]`	Passed
merged lines nest together	`[28, True]`	`[28, True]`	Passed
sku limit	`[14, False]`	`[14, False]`	Passed
capacity equal	`[30, True]`	`[30, True]`	Passed
non nesting	`[80, True]`	`[80, True]`	Passed
generated mix 9	`[374, False]`	`[374, False]`	Passed

SHA-256 / 07b4c772fa4a2221e3b1d5b998be18ea6ac14d11594e88e1bae428f861817e7f

Verification & scope

Stipulated linear nesting model; no orientation or partial stacks. This reproducer isolates one failure mechanism. Results cover the supplied fixtures. Variants within a family share a test contract and should remain grouped when constructing evaluation splits. Related mechanisms with a shared evaluation_group must also remain together; these controlled models are not independent production incidents.

Observations recorded using Python 3.12.14 at 2026-09-29T14:52:11.081547+00:00.

Case digest / 91ebf18c4742169bc337bef3c1ff59ed088df1a6985c7e00bc63ae08c1036441