Overview

3D Bin Packing Formulas for Cold Chain Logistics

This section provides the mathematical foundations for pallet load planning – the formulas, logic, and calculations that determine how mixed box sizes can be arranged on standard pallets. These calculations support the article “Why Palletising Is Harder Than Tetris” and inform our palletisation planning tools.

Understanding these formulas explains why palletisation is genuinely complex, why professional service adds value, and why advance information about box dimensions dramatically improves outcomes.

Pallet Specifications (South African Standard)

Euro Pallet Dimensions

Parameter	Value	Notes
Length	1,200 mm	Four-way forklift entry
Width	1,000 mm
Pallet height	150 mm	Deck thickness
Maximum total height	1,800 mm	Carrier specification
Usable stacking height	1,650 mm	1,800 – 150
Footprint area	1.2 m²	1.0 × 1.2
Usable volume	1,980 L	1.0 × 1.2 × 1.65

Weight Limits by Route

Route	Maximum Gross Weight
Johannesburg ↔ Cape Town	940 kg
Johannesburg ↔ Durban	840 kg
First/last 6 pallets in full load	850 kg
Default (other routes)	850 kg

Combinatorial Complexity

Orientation Combinations

Formula:

O_total = 6^n

Variables:

• O_total = Total possible orientation combinations
• n = Number of boxes
• 6 = Possible orientations per rectangular box (which face up × which face forward)

What It Calculates: The number of ways boxes can be oriented before considering placement. Each rectangular box can be placed with any of its six faces pointing upward, and then rotated around the vertical axis.

Why It Matters: Even before deciding where to place boxes, you’ve already created an exponentially growing problem. For 30 boxes, there are 2.2 × 10²³ orientation combinations – more than the number of stars in the observable universe.

Worked Example:

Given: 30 boxes to palletise

O_total = 6^30
O_total = 2.21 × 10²³

For comparison:
- 10 boxes: 6^10 = 60,466,176 combinations
- 20 boxes: 6^20 = 3.66 × 10¹⁵ combinations
- 30 boxes: 6^30 = 2.21 × 10²³ combinations

Each additional box multiplies complexity by 6.

Placement Sequence Permutations

Formula:

P_total = n!

Variables:

• P_total = Total possible placement sequences
• n = Number of boxes
• n! = n factorial (n × (n-1) × (n-2) × … × 1)

What It Calculates: The number of different orders in which boxes could be placed. Placement order matters because earlier boxes constrain where later boxes can fit.

Why It Matters: Factorial growth is faster than exponential growth. By 30 boxes, you have 2.65 × 10³² possible sequences – a number so large it defies intuition.

Worked Example:

Given: 30 boxes to palletise

P_total = 30!
P_total = 30 × 29 × 28 × ... × 2 × 1
P_total = 2.65 × 10³²

For comparison:
- 10 boxes: 10! = 3,628,800
- 20 boxes: 20! = 2.43 × 10¹⁸
- 30 boxes: 30! = 2.65 × 10³²

Doubling boxes increases permutations by roughly 10^15 (quadrillion-fold).

Total Configuration Space

Formula:

C_total = O_total × P_total = 6^n × n!

Variables:

• C_total = Total possible configurations
• O_total = Orientation combinations
• P_total = Placement permutations

What It Calculates: The complete solution space – every possible way the boxes could be arranged considering both orientation and placement order.

Why It Matters: This is the haystack in which we’re searching for the needle (a good configuration). The vast majority of configurations are physically impossible (boxes overlapping), unstable, or exceed constraints.

Worked Example:

Given: 30 boxes to palletise

C_total = 6^30 × 30!
C_total = 2.21 × 10²³ × 2.65 × 10³²
C_total = 5.86 × 10⁵⁵

For perspective:
- Atoms in Earth: ~10⁵⁰
- Atoms in observable universe: ~10⁸⁰
- 30-box configuration space: 10⁵⁵

The problem space is larger than the number of atoms on Earth.

Computational Implication:

If a computer evaluated 1 billion configurations per second:

Time to exhaustively search = 5.86 × 10⁵⁵ / 10⁹ seconds
                            = 5.86 × 10⁴⁶ seconds
                            = 1.86 × 10³⁹ years

Age of universe: 1.38 × 10¹⁰ years

Exhaustive search would take 10²⁹ times the age of the universe.

This is why 3D bin packing is classified NP-hard – no efficient algorithm exists.

Space Efficiency Calculations

Theoretical Fill Rate

Formula:

η_theoretical = (Σ V_box) / V_pallet × 100%

Where:
V_pallet = L_pallet × W_pallet × H_usable

Variables:

• η_theoretical = Theoretical maximum fill rate (%)
• Σ V_box = Sum of all box volumes (L or m³)
• V_pallet = Usable pallet volume (L or m³)
• L_pallet = Pallet length (1,200 mm)
• W_pallet = Pallet width (1,000 mm)
• H_usable = Usable stacking height (1,650 mm)

What It Calculates: The percentage of pallet volume that could be filled if boxes packed with zero wasted space – the upper bound on efficiency.

Why It Matters: Theoretical fill rate sets expectations. If total box volume is 1,500L on a 1,980L pallet, theoretical maximum is 76%. Actual achieved rates will be lower due to geometric constraints. If theoretical is already below target, you need more pallets regardless of how clever your arrangement is.

Worked Example:

Given: Mixed load
- 8× Stock 3 (9.4 L each) = 75 L
- 10× Stock 5 (40.5 L each) = 405 L
- 6× Stock 6 (81 L each) = 486 L
- 6× Butcher's (30.9 L each) = 185 L
- Total: 1,151 L

V_pallet = 1.0 × 1.2 × 1.65 = 1.98 m³ = 1,980 L

η_theoretical = 1,151 / 1,980 × 100%
η_theoretical = 58.1%

Interpretation: Even perfect packing cannot exceed 58% fill.
Realistic achievable: 45-52% (mixed box sizes prevent perfect tiling).

Single Box Type Layer Efficiency

Formula:

η_layer = (n_L × n_W × A_box) / A_pallet × 100%

Where:
n_L = floor(L_pallet / L_box) or floor(L_pallet / W_box)
n_W = floor(W_pallet / W_box) or floor(W_pallet / L_box)

Variables:

• η_layer = Layer space efficiency (%)
• n_L = Boxes fitting along pallet length
• n_W = Boxes fitting along pallet width
• A_box = Box footprint area (L_box × W_box)
• A_pallet = Pallet footprint area (1,200 × 1,000 mm = 1,200,000 mm²)
• floor() = Round down to nearest integer (partial boxes don’t count)

What It Calculates: How efficiently a single box type covers the pallet footprint in one layer. Tests both orientations to find optimal arrangement.

Why It Matters: Layer efficiency determines how many boxes fit per height level. Low layer efficiency cascades through the entire pallet – you can’t recover lost horizontal space by stacking higher.

Worked Example – Stock 6 (600 × 450 × 300mm):

Orientation A: Length along 1200mm edge
n_L = floor(1200 / 600) = 2
n_W = floor(1000 / 450) = 2
Boxes per layer = 2 × 2 = 4
Coverage = 4 × (600 × 450) = 1,080,000 mm²
η_layer_A = 1,080,000 / 1,200,000 × 100% = 90%

Orientation B: Width along 1200mm edge
n_L = floor(1200 / 450) = 2
n_W = floor(1000 / 600) = 1
Boxes per layer = 2 × 1 = 2
Coverage = 2 × (600 × 450) = 540,000 mm²
η_layer_B = 540,000 / 1,200,000 × 100% = 45%

Optimal: Orientation A (90% layer efficiency, 4 boxes per layer)
Dead space: 1,200,000 - 1,080,000 = 120,000 mm² = 100mm × 1200mm strip

Worked Example – Butcher’s Box (577 × 369 × 145mm):

Orientation A: 577mm along length
n_L = floor(1200 / 577) = 2 (uses 1154mm, leaves 46mm)
n_W = floor(1000 / 369) = 2 (uses 738mm, leaves 262mm)
Boxes per layer = 4
Coverage = 4 × (577 × 369) = 851,652 mm²
η_layer_A = 851,652 / 1,200,000 × 100% = 71%

Orientation B: 369mm along length
n_L = floor(1200 / 369) = 3 (uses 1107mm, leaves 93mm)
n_W = floor(1000 / 577) = 1 (uses 577mm, leaves 423mm)
Boxes per layer = 3
Coverage = 3 × (577 × 369) = 638,739 mm²
η_layer_B = 638,739 / 1,200,000 × 100% = 53%

Optimal: Orientation A (71% layer efficiency)
But 71% is poor - nearly 30% of each layer is wasted.

Layers Per Pallet

Formula:

n_layers = floor(H_usable / H_box)

Variables:

• n_layers = Maximum complete layers
• H_usable = Usable stacking height (1,650 mm)
• H_box = Box height

What It Calculates: How many complete horizontal layers of a given box height fit within the usable pallet height.

Why It Matters: Partial layers are unstable and waste vertical space. If boxes are 300mm tall and you have 1,650mm height, you get 5 layers (1,500mm) with 150mm wasted at top – 9% vertical inefficiency built in.

Worked Example – Standard Stock Boxes:

Stock 3 (250mm): floor(1650/250) = 6 layers, uses 1500mm, wastes 150mm (9%)
Stock 4 (300mm): floor(1650/300) = 5 layers, uses 1500mm, wastes 150mm (9%)
Stock 5 (300mm): floor(1650/300) = 5 layers, uses 1500mm, wastes 150mm (9%)
Stock 6 (300mm): floor(1650/300) = 5 layers, uses 1500mm, wastes 150mm (9%)
Butcher's (145mm): floor(1650/145) = 11 layers, uses 1595mm, wastes 55mm (3%)

Total Boxes Per Pallet (Uniform Load)

Formula:

N_total = n_per_layer × n_layers

Where:
n_per_layer = optimal layer arrangement from efficiency calculation
n_layers = floor(H_usable / H_box)

Variables:

• N_total = Maximum boxes per pallet (uniform product)
• n_per_layer = Boxes per layer (from layer efficiency calculation)
• n_layers = Number of layers

Worked Example – Stock 6:

n_per_layer = 4 (from earlier calculation)
n_layers = 5 (1650/300)
N_total = 4 × 5 = 20 Stock 6 boxes per pallet

Volume check:
20 × 81 L = 1,620 L product
1,620 / 1,980 = 81.8% volume utilisation

But layer efficiency was only 90%, so:
Actual space used = 90% × (1500/1650) × 100% = 81.8% ✓

Worked Example – Butcher’s Boxes:

n_per_layer = 4 (from earlier calculation)
n_layers = 11 (1650/145)
N_total = 4 × 11 = 44 butcher's boxes per pallet

Volume check:
44 × 30.9 L = 1,360 L product
1,360 / 1,980 = 68.7% volume utilisation

Layer efficiency 71% × vertical efficiency 97% = 68.9% ≈ 68.7% ✓

Weight Distribution

Centre of Gravity Height

Formula:

H_cg = Σ(m_i × h_i) / Σ(m_i)

Variables:

• H_cg = Height of centre of gravity above pallet deck (mm)
• m_i = Mass of each box or layer (kg)
• h_i = Height of each box’s centre from pallet deck (mm)

What It Calculates: The vertical position of the pallet’s centre of mass. Lower is more stable.

Why It Matters: High centre of gravity increases tip-over risk during forklift handling and cornering. Target: H_cg below 50% of total height (below 825mm for 1,650mm stack).

Worked Example – Heavy Bottom vs Heavy Top:

Scenario: 5 layers, each 100kg, total height 1500mm

Layer centres from deck:
Layer 1: 150mm (centre of 0-300mm)
Layer 2: 450mm
Layer 3: 750mm
Layer 4: 1050mm
Layer 5: 1350mm

Heavy-at-bottom (Layer 1 = 200kg, others = 75kg):
H_cg = (200×150 + 75×450 + 75×750 + 75×1050 + 75×1350) / 500
H_cg = (30000 + 33750 + 56250 + 78750 + 101250) / 500
H_cg = 300,000 / 500 = 600mm ✓ (below 50%)

Heavy-at-top (Layer 5 = 200kg, others = 75kg):
H_cg = (75×150 + 75×450 + 75×750 + 75×1050 + 200×1350) / 500
H_cg = (11250 + 33750 + 56250 + 78750 + 270000) / 500
H_cg = 450,000 / 500 = 900mm ✗ (above 50%, unstable)

Crush Weight Per Box

Formula:

W_crush = Σ(m_above) × g × SF

Where:
m_above = total mass of all boxes stacked above the box in question
SF = safety factor (typically 1.5-2.0 for dynamic loads)

Variables:

• W_crush = Effective crush weight on box (N or kg-force)
• m_above = Mass above the box (kg)
• g = Gravitational acceleration (9.81 m/s², often simplified to 10)
• SF = Safety factor for dynamic loads (vibration, braking, impacts)

What It Calculates: The compressive force experienced by boxes at each layer, accounting for transport dynamics.

Why It Matters: Cardboard boxes have crush ratings (burst test values). Exceeding crush rating damages product and collapses the stack. Bottom layer boxes experience the most stress.

Worked Example:

5-layer stack, each layer 100kg, SF = 1.5 for road transport

Layer 5 (top): W_crush = 0 × 1.5 = 0 kg-force
Layer 4: W_crush = 100 × 1.5 = 150 kg-force
Layer 3: W_crush = 200 × 1.5 = 300 kg-force
Layer 2: W_crush = 300 × 1.5 = 450 kg-force
Layer 1 (bottom): W_crush = 400 × 1.5 = 600 kg-force

If bottom layer has 4 boxes:
Per-box crush load = 600 / 4 = 150 kg-force per box

Stock 6 cartons typically rated for 100-150kg stacking weight.
At 150 kg-force with SF 1.5, boxes are at limit.
Add humidity (weakens cardboard) or shock events → failure risk.

Mixed Load Complexity

Height Mismatch Penalty

Formula:

η_mixed = Σ(n_i × V_i) / V_used

Where:
V_used = A_pallet × max(H_tallest_stack)

Variables:

• η_mixed = Actual efficiency of mixed-height arrangement
• n_i = Quantity of each box type
• V_i = Volume of each box type
• V_used = Volume of pallet space consumed by the arrangement

What It Calculates: The effective space utilisation when boxes of different heights are combined, accounting for the “tallest stack wins” problem.

Why It Matters: When mixing heights, the tallest stack determines the layer height. Shorter boxes in that layer waste vertical space. Two 145mm butcher’s boxes (290mm total) next to one 300mm Stock 5 waste 10mm. Over 5 layers, that’s 50mm × footprint area of dead space.

Worked Example:

Layer mixing Stock 5 (300mm) and Butcher's (145mm):

Option A: Separate layers
- Stock 5: 4 boxes × 300mm layers
- Butcher's: 4 boxes × 145mm layers
- Both tile independently at their optimal efficiencies

Option B: Same layer
- Must use 300mm layer height (taller box wins)
- Butcher's boxes waste 300 - 145 = 155mm vertical each
- Or stack 2 butcher's (290mm) leaving 10mm gap

Penalty calculation for mixed layer:
2 butcher's stacked = 290mm, in 300mm layer space
Wasted = 10mm per stack position
If 2 stack positions per layer, 5 layers:
Total waste = 10mm × 2 × 5 = 100mm × footprint area

For butcher's footprint (~577 × 369mm):
Wasted volume = 0.577 × 0.369 × 0.1 × 2 = 0.043 m³ = 43 L

Practical Heuristics

Layer Building Priority

Logic:

1. Sort boxes by height (descending)
2. Group boxes within ±10mm height tolerance
3. Build complete layers from each height group
4. Stack layer groups with tallest at bottom

Why It Works: Minimises height mismatch penalties by keeping similar heights together. Tall layers at bottom places heavy items low (assuming larger boxes are heavier).

Gap Filling Strategy

Logic:

1. Calculate residual space after major box placement
2. If gap > smallest box dimension: attempt to fill
3. If gap < smallest box dimension: accept as dead space
4. Position dead space toward pallet centre, not edges

Why It Works: Centre gaps get compressed by surrounding load. Edge gaps allow shifting and collapse. Small gaps that can’t be filled aren’t worth solving – move to the next pallet.

Weight Limit Check

Formula:

Pass/Fail = (Σ m_boxes + m_pallet + m_wrap) ≤ W_limit_route

Typical Values:

• m_pallet = 20-25 kg (wooden Euro pallet)
• m_wrap = 1-2 kg (shrink wrap)
• W_limit_route = 840-940 kg depending on destination

Application: Before finalising arrangement, verify total weight. Space-optimal arrangements may exceed weight limits, requiring removal of bottom-layer boxes (the heavy ones contributing to stability).

Tool Implementation Logic

Input Requirements

For accurate palletisation planning, the tool requires:

Input	Format	Purpose
Box dimensions	L × W × H (mm)	Tiling calculations
Quantities	Integer per box type	Volume and count totals
Box weight	kg per box	Weight limit and distribution
Crush rating	kg (optional)	Stacking constraints
Route	JHB↔CPT / JHB↔DBN / Other	Weight limit lookup

Output Calculations

1. Theoretical fill rate – Upper bound on efficiency
2. Layer efficiency by box type – Optimal orientation for each
3. Estimated pallet count – Ceiling of (total volume / achievable pallet volume)
4. Weight distribution preview – Centre of gravity estimate
5. Constraint warnings – Weight limit, crush risk, stability concerns

Algorithm Approach

Given NP-hard complexity, the tool uses heuristic approximation:

1. Sort boxes by volume (descending) - large boxes first
2. For each box type, calculate optimal layer orientation
3. Build virtual layers, grouping by compatible heights
4. Stack layers bottom-to-top, heavy first
5. Calculate residual space, attempt gap filling with small boxes
6. Verify constraints (weight, stability, crush)
7. Output arrangement and pallet count estimate

This produces good solutions (typically within 10-15% of optimal) in milliseconds rather than millennia.

Validation Notes

These formulas represent:

• Standard 3D bin packing mathematics
• Practical constraints from South African long-haul requirements
• Operational experience from The Frozen Food Courier

Actual results vary based on:

• Box condition (warped cardboard reduces tiling efficiency)
• Product-specific constraints (orientation requirements, crush sensitivity)
• Carrier-specific rules (some require specific label positions affecting orientation)

Related Resources:

• Palletising Article
• Palletisation Guide
• Technical Formulas Reference – Main Page

Mathematics doesn’t make palletising easy. It explains why it’s hard – and why getting it right matters.