FA-95151 / Warehouse slotting and bin packing / Open access

Nested bin fill: first unit counted at nest increment · case 01

Nested stacks are credited far less volume than they occupy, overfilling bins.

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

ROOT CAUSE

Every nested unit is sized at the nest increment, dropping the full volume of the base unit.

VERIFIED REPAIR

Size a nested stack as one full unit plus (qty - 1) increments.

Unsuccessful approach: Adding qty increments on top of the base unit double counts the base.

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():
        if qty <= 0:
            continue
        count += 1
        if nv > 0:
            total += qty * 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 = [[['nesting', [[['CUP', 10, 10, 2]], 30, 1], [28, True]], ['capacity equal', [[['CUP', 11, 10, 2]], 30, 1], [30, 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]], ['zero line', [[['CUP', 1, 10, 2], ['BOX', 0, 40, 0]], 30, 1], [10, True]], ['generated mix 7', [[['TRY', 2, 25, 5]], 120, 2], [30, True]], ['generated mix 15', [[['CUP', 2, 10, 2], ['BWL', 1, 30, 6], ['BOX', 2, 40, 0], ['BWL', 1, 30, 6]], 150, 3], [128, True]], ['generated mix 23', [[['TRY', 2, 25, 5], ['CUP', 10, 10, 2], ['TRY', 0, 25, 5]], 150, 2], [58, True]]], [['generated mix 3', [[['CUP', 5, 10, 2], ['TRY', 0, 25, 5], ['LID', 0, 4, 0]], 100, 3], [18, True]], ['generated mix 2', [[['BWL', 2, 30, 6], ['LID', 0, 4, 0], ['BWL', 1, 30, 6]], 120, 3], [42, 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 11', [[['BWL', 1, 30, 6], ['LID', 2, 4, 0], ['LID', 0, 4, 0], ['LID', 5, 4, 0], ['BOX', 5, 40, 0]], 120, 3], [258, False]], ['generated mix 19', [[['BOX', 2, 40, 0], ['TRY', 0, 25, 5]], 80, 3], [80, True]], ['generated mix 27', [[['CUP', 2, 10, 2]], 100, 1], [12, True]]], [['generated mix 11', [[['BWL', 1, 30, 6], ['LID', 2, 4, 0], ['LID', 0, 4, 0], ['LID', 5, 4, 0], ['BOX', 5, 40, 0]], 120, 3], [258, False]], ['generated mix 7', [[['TRY', 2, 25, 5]], 120, 2], [30, 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 22', [[['BOX', 2, 40, 0], ['BOX', 10, 40, 0], ['BOX', 0, 40, 0], ['BOX', 2, 40, 0], ['LID', 1, 4, 0]], 50, 2], [564, False]], ['generated mix 30', [[['BOX', 1, 40, 0], ['TRY', 2, 25, 5]], 120, 1], [70, False]], ['generated mix 38', [[['BWL', 1, 30, 6], ['BOX', 2, 40, 0], ['LID', 5, 4, 0]], 120, 1], [130, False]]], [['generated mix 23', [[['TRY', 2, 25, 5], ['CUP', 10, 10, 2], ['TRY', 0, 25, 5]], 150, 2], [58, True]], ['generated mix 15', [[['CUP', 2, 10, 2], ['BWL', 1, 30, 6], ['BOX', 2, 40, 0], ['BWL', 1, 30, 6]], 150, 3], [128, 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 34', [[['BWL', 10, 30, 6], ['BOX', 1, 40, 0], ['TRY', 2, 25, 5], ['LID', 10, 4, 0], ['BOX', 10, 40, 0]], 100, 1], [594, False]], ['generated mix 23', [[['TRY', 2, 25, 5], ['CUP', 10, 10, 2], ['TRY', 0, 25, 5]], 150, 2], [58, 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 44', [[['CUP', 10, 10, 2]], 150, 1], [28, 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
nesting	`[20, True]`	`[28, True]`	Failed
capacity equal	`[22, True]`	`[30, True]`	Failed
merged lines nest together	`[20, True]`	`[28, True]`	Failed
non nesting	`[80, True]`	`[80, True]`	Passed
zero line	`[2, True]`	`[10, True]`	Failed
generated mix 7	`[10, True]`	`[30, True]`	Failed
generated mix 15	`[96, True]`	`[128, True]`	Failed
generated mix 23	`[30, True]`	`[58, True]`	Failed

SHA-256 / 40bda4df5484e8f15cbb9382eaf0cb5be54eb7129ca54800f8f018edbf1608df

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():
        if qty <= 0:
            continue
        count += 1
        if nv > 0:
            total += uv + qty * 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 = [[['nesting', [[['CUP', 10, 10, 2]], 30, 1], [28, True]], ['capacity equal', [[['CUP', 11, 10, 2]], 30, 1], [30, 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]], ['zero line', [[['CUP', 1, 10, 2], ['BOX', 0, 40, 0]], 30, 1], [10, True]], ['generated mix 7', [[['TRY', 2, 25, 5]], 120, 2], [30, True]], ['generated mix 15', [[['CUP', 2, 10, 2], ['BWL', 1, 30, 6], ['BOX', 2, 40, 0], ['BWL', 1, 30, 6]], 150, 3], [128, True]], ['generated mix 23', [[['TRY', 2, 25, 5], ['CUP', 10, 10, 2], ['TRY', 0, 25, 5]], 150, 2], [58, True]]], [['generated mix 3', [[['CUP', 5, 10, 2], ['TRY', 0, 25, 5], ['LID', 0, 4, 0]], 100, 3], [18, True]], ['generated mix 2', [[['BWL', 2, 30, 6], ['LID', 0, 4, 0], ['BWL', 1, 30, 6]], 120, 3], [42, 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 11', [[['BWL', 1, 30, 6], ['LID', 2, 4, 0], ['LID', 0, 4, 0], ['LID', 5, 4, 0], ['BOX', 5, 40, 0]], 120, 3], [258, False]], ['generated mix 19', [[['BOX', 2, 40, 0], ['TRY', 0, 25, 5]], 80, 3], [80, True]], ['generated mix 27', [[['CUP', 2, 10, 2]], 100, 1], [12, True]]], [['generated mix 11', [[['BWL', 1, 30, 6], ['LID', 2, 4, 0], ['LID', 0, 4, 0], ['LID', 5, 4, 0], ['BOX', 5, 40, 0]], 120, 3], [258, False]], ['generated mix 7', [[['TRY', 2, 25, 5]], 120, 2], [30, 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 22', [[['BOX', 2, 40, 0], ['BOX', 10, 40, 0], ['BOX', 0, 40, 0], ['BOX', 2, 40, 0], ['LID', 1, 4, 0]], 50, 2], [564, False]], ['generated mix 30', [[['BOX', 1, 40, 0], ['TRY', 2, 25, 5]], 120, 1], [70, False]], ['generated mix 38', [[['BWL', 1, 30, 6], ['BOX', 2, 40, 0], ['LID', 5, 4, 0]], 120, 1], [130, False]]], [['generated mix 23', [[['TRY', 2, 25, 5], ['CUP', 10, 10, 2], ['TRY', 0, 25, 5]], 150, 2], [58, True]], ['generated mix 15', [[['CUP', 2, 10, 2], ['BWL', 1, 30, 6], ['BOX', 2, 40, 0], ['BWL', 1, 30, 6]], 150, 3], [128, 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 34', [[['BWL', 10, 30, 6], ['BOX', 1, 40, 0], ['TRY', 2, 25, 5], ['LID', 10, 4, 0], ['BOX', 10, 40, 0]], 100, 1], [594, False]], ['generated mix 23', [[['TRY', 2, 25, 5], ['CUP', 10, 10, 2], ['TRY', 0, 25, 5]], 150, 2], [58, 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 44', [[['CUP', 10, 10, 2]], 150, 1], [28, 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
nesting	`[30, True]`	`[28, True]`	Failed
capacity equal	`[32, False]`	`[30, True]`	Failed
merged lines nest together	`[30, True]`	`[28, True]`	Failed
non nesting	`[80, True]`	`[80, True]`	Passed
zero line	`[12, True]`	`[10, True]`	Failed
generated mix 7	`[35, True]`	`[30, True]`	Failed
generated mix 15	`[136, True]`	`[128, True]`	Failed
generated mix 23	`[65, True]`	`[58, True]`	Failed

SHA-256 / 8b9f4bc035aa13e50c720930b6a7f3cf2d4040ebad71ab5057a579827f181af2

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 = [[['nesting', [[['CUP', 10, 10, 2]], 30, 1], [28, True]], ['capacity equal', [[['CUP', 11, 10, 2]], 30, 1], [30, 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]], ['zero line', [[['CUP', 1, 10, 2], ['BOX', 0, 40, 0]], 30, 1], [10, True]], ['generated mix 7', [[['TRY', 2, 25, 5]], 120, 2], [30, True]], ['generated mix 15', [[['CUP', 2, 10, 2], ['BWL', 1, 30, 6], ['BOX', 2, 40, 0], ['BWL', 1, 30, 6]], 150, 3], [128, True]], ['generated mix 23', [[['TRY', 2, 25, 5], ['CUP', 10, 10, 2], ['TRY', 0, 25, 5]], 150, 2], [58, True]]], [['generated mix 3', [[['CUP', 5, 10, 2], ['TRY', 0, 25, 5], ['LID', 0, 4, 0]], 100, 3], [18, True]], ['generated mix 2', [[['BWL', 2, 30, 6], ['LID', 0, 4, 0], ['BWL', 1, 30, 6]], 120, 3], [42, 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 11', [[['BWL', 1, 30, 6], ['LID', 2, 4, 0], ['LID', 0, 4, 0], ['LID', 5, 4, 0], ['BOX', 5, 40, 0]], 120, 3], [258, False]], ['generated mix 19', [[['BOX', 2, 40, 0], ['TRY', 0, 25, 5]], 80, 3], [80, True]], ['generated mix 27', [[['CUP', 2, 10, 2]], 100, 1], [12, True]]], [['generated mix 11', [[['BWL', 1, 30, 6], ['LID', 2, 4, 0], ['LID', 0, 4, 0], ['LID', 5, 4, 0], ['BOX', 5, 40, 0]], 120, 3], [258, False]], ['generated mix 7', [[['TRY', 2, 25, 5]], 120, 2], [30, 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 22', [[['BOX', 2, 40, 0], ['BOX', 10, 40, 0], ['BOX', 0, 40, 0], ['BOX', 2, 40, 0], ['LID', 1, 4, 0]], 50, 2], [564, False]], ['generated mix 30', [[['BOX', 1, 40, 0], ['TRY', 2, 25, 5]], 120, 1], [70, False]], ['generated mix 38', [[['BWL', 1, 30, 6], ['BOX', 2, 40, 0], ['LID', 5, 4, 0]], 120, 1], [130, False]]], [['generated mix 23', [[['TRY', 2, 25, 5], ['CUP', 10, 10, 2], ['TRY', 0, 25, 5]], 150, 2], [58, True]], ['generated mix 15', [[['CUP', 2, 10, 2], ['BWL', 1, 30, 6], ['BOX', 2, 40, 0], ['BWL', 1, 30, 6]], 150, 3], [128, 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 34', [[['BWL', 10, 30, 6], ['BOX', 1, 40, 0], ['TRY', 2, 25, 5], ['LID', 10, 4, 0], ['BOX', 10, 40, 0]], 100, 1], [594, False]], ['generated mix 23', [[['TRY', 2, 25, 5], ['CUP', 10, 10, 2], ['TRY', 0, 25, 5]], 150, 2], [58, 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 44', [[['CUP', 10, 10, 2]], 150, 1], [28, 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
nesting	`[28, True]`	`[28, True]`	Passed
capacity equal	`[30, True]`	`[30, True]`	Passed
merged lines nest together	`[28, True]`	`[28, True]`	Passed
non nesting	`[80, True]`	`[80, True]`	Passed
zero line	`[10, True]`	`[10, True]`	Passed
generated mix 7	`[30, True]`	`[30, True]`	Passed
generated mix 15	`[128, True]`	`[128, True]`	Passed
generated mix 23	`[58, True]`	`[58, True]`	Passed

SHA-256 / ca198e9d586b0622c926aabbf12bf6f20977f1aefa66a6142ace6bf2d2899fd6

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 2026-09-29T14:52:11.077831+00:00.

Case digest / abde21fdba06f7366423bd5630f7b576fd935cc3fc8a9e1918fa0ea55a404948