pallet-loading（托盘装载）

Pallet Loading Optimization

You are an expert in pallet loading and optimization. Your goal is to help efficiently load boxes, cases, and items onto pallets while maximizing utilization, ensuring load stability, meeting weight constraints, and complying with shipping requirements.

Initial Assessment

Before optimizing pallet loading, understand:

1. Pallet Specifications

- Pallet type? (GMA/CHEP 48x40, Euro 47.2x39.4, custom) - Pallet dimensions: length x width x height - Pallet weight capacity? (typically 2,500-4,600 lbs) - Pallet weight itself? (typically 30-70 lbs) - Overhang allowed? (typically 0-3 inches)

1. Items to Load

- How many items/cases? What dimensions? - Item weights and weight distribution - Stackability constraints? (stack limit, crush strength) - Fragile items requiring special placement? - All same size or mixed SKUs?

1. Load Constraints

- Maximum pallet height? (often 72", 84", 96") - Weight distribution requirements? - Interlocking pattern required? - Stretch wrap, shrink wrap, or banding? - Anti-slip sheets needed?

1. Shipping Requirements

- Truck or rail shipment? - Will pallets be stacked? (double-stacking) - Fork lift access directions? - Loading dock height constraints? - Transit duration and handling?

1. Optimization Goals

- Maximize cases per pallet? - Minimize number of pallets? - Ensure stability (minimize overhang)? - Optimize for warehouse space?

Pallet Loading Framework

Standard Pallet Sizes

North America:

• GMA/CHEP: 48" x 40" (1219mm x 1016mm) - Most common
• 42" x 42" - Telecommunications, paint
• 48" x 48" - Drums, bulk materials
• 40" x 48" - Military, cement

Europe:

• EUR/EUR1: 47.2" x 31.5" (1200mm x 800mm) - European standard
• EUR2: 47.2" x 23.6" (1200mm x 600mm)
• EUR3: 39.4" x 47.2" (1000mm x 1200mm)
• EUR6: 31.5" x 47.2" (800mm x 1200mm)

Asia:

• 43.3" x 43.3" (1100mm x 1100mm) - Asian standard
• 39.4" x 39.4" (1000mm x 1000mm)

ISO Standards:

• ISO 1: 47.2" x 39.4" (1200mm x 1000mm)
• ISO 2: 39.4" x 47.2" (1000mm x 1200mm)

Key Concepts

1. Pallet Pattern

• Arrangement of boxes on each layer
• Determines stability and efficiency
• Common patterns: block, row, pinwheel, split-row

2. Interlocking

• Boxes overlap between layers (90° rotation)
• Improves stability
• Reduces risk of collapse

3. Column Stacking

• Boxes directly on top of each other
• No interlocking
• Less stable but sometimes required

4. Overhang

• Box extends beyond pallet edge
• Typically limited to 2-3 inches
• Reduces stability, increases damage risk

5. Compression Strength

• Weight box can support from above
• Critical for determining stack height
• Degrades over time and with humidity

Mathematical Formulation

Pallet Loading Problem

Objective: Maximize number of boxes loaded OR minimize number of pallets

Variables:

• x_i, y_i = position of box i on pallet
• z_i = layer number for box i
• o_i = orientation of box i (0-5)
• pattern_j = pallet pattern used for layer j

Constraints:

1. Pallet dimensions: boxes fit within pallet bounds (with overhang tolerance)
2. Non-overlap: boxes don't overlap in same layer
3. Weight: total weight ≤ pallet capacity
4. Stability: adequate support from below, center of gravity
5. Stack height: height ≤ maximum allowed
6. Compression: weight on each box ≤ its compression strength

Algorithms and Solution Methods

Pattern Generation Methods

Simple Block Pattern

def generate_block_pattern(box_dims, pallet_dims, allow_overhang=2):
    """
    Generate simple block pattern for single box size

    Calculates optimal orientation and layout

    Parameters:
    - box_dims: (length, width, height)
    - pallet_dims: (length, width, max_height)
    - allow_overhang: maximum overhang in inches

    Returns pattern with max boxes per layer
    """

    box_l, box_w, box_h = box_dims
    pallet_l, pallet_w, pallet_h = pallet_dims

    best_pattern = None
    max_boxes = 0

    # Try both orientations
    orientations = [
        (box_l, box_w, box_h, 0),  # Original
        (box_w, box_l, box_h, 1)   # Rotated 90°
    ]

    for l, w, h, rotation in orientations:
        # Calculate how many fit in each direction
        effective_pallet_l = pallet_l + 2 * allow_overhang
        effective_pallet_w = pallet_w + 2 * allow_overhang

        boxes_length = int(effective_pallet_l // l)
        boxes_width = int(effective_pallet_w // w)

        boxes_per_layer = boxes_length * boxes_width

        if boxes_per_layer > max_boxes:
            max_boxes = boxes_per_layer

            # Calculate actual overhang
            actual_overhang_l = (boxes_length * l - pallet_l) / 2
            actual_overhang_w = (boxes_width * w - pallet_w) / 2

            best_pattern = {
                'boxes_per_layer': boxes_per_layer,
                'boxes_length': boxes_length,
                'boxes_width': boxes_width,
                'rotation': rotation,
                'box_dims': (l, w, h),
                'overhang': (actual_overhang_l, actual_overhang_w),
                'positions': []
            }

            # Generate positions
            for i in range(boxes_length):
                for j in range(boxes_width):
                    x = i * l - actual_overhang_l
                    y = j * w - actual_overhang_w
                    best_pattern['positions'].append((x, y, l, w))

    return best_pattern

# Example
box = (12, 10, 8)  # inches
pallet = (48, 40, 72)  # GMA pallet

pattern = generate_block_pattern(box, pallet, allow_overhang=2)
print(f"Boxes per layer: {pattern['boxes_per_layer']}")
print(f"Layout: {pattern['boxes_length']} x {pattern['boxes_width']}")
print(f"Overhang: {pattern['overhang']}")

Interlocking Pattern Generator

def generate_interlocking_pattern(box_dims, pallet_dims, num_layers):
    """
    Generate interlocking pattern for stability

    Alternates orientation between layers

    Parameters:
    - box_dims: (length, width, height)
    - pallet_dims: (length, width, max_height)
    - num_layers: number of layers to generate

    Returns multi-layer interlocking pattern
    """

    box_l, box_w, box_h = box_dims
    pallet_l, pallet_w, pallet_h = pallet_dims

    patterns = []

    for layer in range(num_layers):
        # Alternate orientation by layer
        if layer % 2 == 0:
            # Layer 0, 2, 4, ... : original orientation
            pattern = generate_block_pattern(
                (box_l, box_w, box_h),
                (pallet_l, pallet_w, pallet_h)
            )
        else:
            # Layer 1, 3, 5, ... : rotated orientation
            pattern = generate_block_pattern(
                (box_w, box_l, box_h),
                (pallet_l, pallet_w, pallet_h)
            )

        # Add z-coordinate (layer height)
        z = layer * box_h
        pattern['layer'] = layer
        pattern['z'] = z

        patterns.append(pattern)

    total_boxes = sum(p['boxes_per_layer'] for p in patterns)

    return {
        'num_layers': num_layers,
        'total_boxes': total_boxes,
        'total_height': num_layers * box_h,
        'layers': patterns
    }

# Example
box = (16, 12, 10)
pallet = (48, 40, 72)
max_layers = int(pallet[2] // box[2])  # 7 layers fit

interlock = generate_interlocking_pattern(box, pallet, max_layers)
print(f"Total boxes: {interlock['total_boxes']}")
print(f"Height: {interlock['total_height']} inches")

Mixed SKU Pattern Optimization

from pulp import *

def optimize_mixed_sku_pallet(items, pallet_dims, max_weight):
    """
    Optimize pallet loading with mixed SKUs

    Uses integer programming to select best combination

    Parameters:
    - items: list of dicts with 'id', 'dims', 'weight', 'quantity'
    - pallet_dims: (length, width, max_height)
    - max_weight: weight capacity

    Returns optimal loading pattern
    """

    pallet_l, pallet_w, pallet_h = pallet_dims

    # Pre-calculate patterns for each item type
    item_patterns = {}
    for item in items:
        item_id = item['id']
        dims = item['dims']

        pattern = generate_block_pattern(dims, pallet_dims)
        item_patterns[item_id] = pattern

    # Create optimization problem
    prob = LpProblem("Pallet_Loading", LpMaximize)

    # Decision variables: how many of each item to load
    x = LpVariable.dicts("load",
                        [item['id'] for item in items],
                        lowBound=0,
                        cat='Integer')

    # Objective: maximize total value/volume loaded
    prob += lpSum([
        x[item['id']] * item['dims'][0] * item['dims'][1] * item['dims'][2]
        for item in items
    ]), "Total_Volume"

    # Weight constraint
    prob += lpSum([
        x[item['id']] * item['weight']
        for item in items
    ]) <= max_weight, "Weight_Capacity"

    # Quantity constraint
    for item in items:
        prob += x[item['id']] <= item['quantity'], f"Quantity_{item['id']}"

    # Height constraint (simplified - assumes stacking)
    # This is a simplification; full solution needs layer-by-layer check
    prob += lpSum([
        x[item['id']] * item['dims'][2]
        for item in items
    ]) <= pallet_h * 10, "Height_Limit"  # Rough approximation

    # Solve
    prob.solve(PULP_CBC_CMD(msg=0))

    # Extract solution
    solution = {
        'status': LpStatus[prob.status],
        'items_loaded': {},
        'total_weight': 0,
        'total_volume': 0
    }

    for item in items:
        qty = int(x[item['id']].varValue) if x[item['id']].varValue else 0
        if qty > 0:
            solution['items_loaded'][item['id']] = qty
            solution['total_weight'] += qty * item['weight']
            volume = item['dims'][0] * item['dims'][1] * item['dims'][2]
            solution['total_volume'] += qty * volume

    return solution

Stability Analysis

Center of Gravity Check

import numpy as np

def check_center_of_gravity(loaded_items, pallet_dims):
    """
    Calculate center of gravity and check stability

    Parameters:
    - loaded_items: list of dicts with 'position' (x,y,z), 'dims', 'weight'
    - pallet_dims: (length, width, height)

    Returns COG coordinates and stability score
    """

    if not loaded_items:
        return None

    total_weight = sum(item['weight'] for item in loaded_items)

    # Calculate weighted average position
    cog_x = sum(
        (item['position'][0] + item['dims'][0]/2) * item['weight']
        for item in loaded_items
    ) / total_weight

    cog_y = sum(
        (item['position'][1] + item['dims'][1]/2) * item['weight']
        for item in loaded_items
    ) / total_weight

    cog_z = sum(
        (item['position'][2] + item['dims'][2]/2) * item['weight']
        for item in loaded_items
    ) / total_weight

    # Check if COG is within pallet bounds
    pallet_center_x = pallet_dims[0] / 2
    pallet_center_y = pallet_dims[1] / 2

    offset_x = abs(cog_x - pallet_center_x)
    offset_y = abs(cog_y - pallet_center_y)

    # Stability score (lower offset = higher stability)
    # Good: offset < 10% of pallet dimension
    max_offset_x = pallet_dims[0] * 0.1
    max_offset_y = pallet_dims[1] * 0.1

    stability_x = max(0, 100 * (1 - offset_x / max_offset_x))
    stability_y = max(0, 100 * (1 - offset_y / max_offset_y))
    stability_score = min(stability_x, stability_y)

    return {
        'cog': (cog_x, cog_y, cog_z),
        'pallet_center': (pallet_center_x, pallet_center_y),
        'offset': (offset_x, offset_y),
        'stability_score': stability_score,
        'is_stable': stability_score > 70
    }

# Example
items = [
    {'position': (0, 0, 0), 'dims': (12, 10, 8), 'weight': 25},
    {'position': (12, 0, 0), 'dims': (12, 10, 8), 'weight': 25},
    {'position': (0, 10, 0), 'dims': (12, 10, 8), 'weight': 25},
    {'position': (12, 10, 0), 'dims': (12, 10, 8), 'weight': 25},
]

pallet = (48, 40, 72)
cog_result = check_center_of_gravity(items, pallet)
print(f"COG: {cog_result['cog']}")
print(f"Stability Score: {cog_result['stability_score']:.1f}")
print(f"Stable: {cog_result['is_stable']}")

Compression Strength Check

def check_compression_strength(layers, box_compression_strength):
    """
    Verify each box can support weight above it

    Parameters:
    - layers: list of layer patterns (bottom to top)
    - box_compression_strength: max weight box can support (lbs)

    Returns safety analysis
    """

    issues = []
    safe = True

    for layer_idx in range(len(layers) - 1):
        layer = layers[layer_idx]

        # Calculate weight above this layer
        weight_above = 0
        for upper_layer_idx in range(layer_idx + 1, len(layers)):
            upper_layer = layers[upper_layer_idx]
            weight_above += sum(
                item['weight'] for item in upper_layer['items']
            )

        # Check each box in this layer
        num_boxes_in_layer = len(layer['items'])
        weight_per_box = weight_above / num_boxes_in_layer if num_boxes_in_layer > 0 else 0

        if weight_per_box > box_compression_strength:
            issues.append({
                'layer': layer_idx,
                'weight_per_box': weight_per_box,
                'max_allowed': box_compression_strength,
                'safety_factor': weight_per_box / box_compression_strength
            })
            safe = False

    return {
        'safe': safe,
        'issues': issues,
        'max_safe_layers': len(layers) if safe else layer_idx
    }

Complete Pallet Optimizer

import matplotlib.pyplot as plt
import matplotlib.patches as patches
from mpl_toolkits.mplot3d import Axes3D
from mpl_toolkits.mplot3d.art3d import Poly3DCollection
import numpy as np

class PalletOptimizer:
    """
    Comprehensive Pallet Loading Optimizer

    Handles single and mixed SKU loading with stability checks
    """

    def __init__(self, pallet_type='GMA', custom_dims=None,
                 weight_capacity=2500, max_height=72):
        """
        Initialize pallet optimizer

        Parameters:
        - pallet_type: 'GMA', 'EUR', 'EUR1', or 'custom'
        - custom_dims: (length, width) if pallet_type='custom'
        - weight_capacity: max weight in lbs
        - max_height: max height in inches
        """

        # Standard pallet dimensions (inches)
        pallet_dims = {
            'GMA': (48, 40),
            'CHEP': (48, 40),
            'EUR': (47.2, 31.5),
            'EUR1': (47.2, 31.5),
            'EUR2': (47.2, 23.6),
            'ISO1': (47.2, 39.4),
            'ISO2': (39.4, 47.2)
        }

        if pallet_type == 'custom':
            if custom_dims is None:
                raise ValueError("Must provide custom_dims for custom pallet")
            self.pallet_dims = (*custom_dims, max_height)
        else:
            dims = pallet_dims.get(pallet_type, (48, 40))
            self.pallet_dims = (*dims, max_height)

        self.pallet_type = pallet_type
        self.weight_capacity = weight_capacity
        self.items = []
        self.solution = None

    def add_item(self, length, width, height, weight,
                 quantity=1, item_id=None, compression_strength=None,
                 fragile=False):
        """Add item type to be loaded"""

        if item_id is None:
            item_id = f"SKU_{len(self.items)}"

        # Default compression strength (stack limit)
        if compression_strength is None:
            compression_strength = weight * 10  # Conservative estimate

        self.items.append({
            'id': item_id,
            'dims': (length, width, height),
            'weight': weight,
            'quantity': quantity,
            'compression_strength': compression_strength,
            'fragile': fragile
        })

    def optimize_single_sku(self, allow_overhang=2, interlock=True):
        """
        Optimize loading for single SKU

        Parameters:
        - allow_overhang: max overhang in inches
        - interlock: use interlocking pattern

        Returns optimal pallet configuration
        """

        if len(self.items) != 1:
            raise ValueError("Use optimize_single_sku only for single SKU")

        item = self.items[0]
        box_dims = item['dims']
        box_weight = item['weight']
        available_qty = item['quantity']

        # Calculate max layers by height
        max_layers_height = int(self.pallet_dims[2] // box_dims[2])

        # Calculate max layers by weight
        max_layers_weight = int(self.weight_capacity //
                               (box_weight * 100))  # rough estimate

        # Calculate max layers by compression
        max_layers_compression = int(item['compression_strength'] / box_weight) + 1

        # Take minimum
        max_layers = min(max_layers_height, max_layers_weight,
                        max_layers_compression, 15)  # Cap at 15 for safety

        if interlock:
            pattern = generate_interlocking_pattern(
                box_dims, self.pallet_dims, max_layers
            )
        else:
            # Simple block stacking
            base_pattern = generate_block_pattern(
                box_dims, self.pallet_dims, allow_overhang
            )

            pattern = {
                'num_layers': max_layers,
                'total_boxes': base_pattern['boxes_per_layer'] * max_layers,
                'layers': [base_pattern] * max_layers
            }

        # Check if we have enough quantity
        if pattern['total_boxes'] > available_qty:
            # Reduce layers to fit available quantity
            boxes_per_layer = pattern['layers'][0]['boxes_per_layer']
            actual_layers = int(available_qty // boxes_per_layer)
            pattern['num_layers'] = actual_layers
            pattern['total_boxes'] = boxes_per_layer * actual_layers

        # Calculate weight
        total_weight = pattern['total_boxes'] * box_weight

        # Check stability
        loaded_items = []
        for layer_idx in range(pattern['num_layers']):
            z = layer_idx * box_dims[2]
            layer_pattern = pattern['layers'][layer_idx % len(pattern['layers'])]

            for pos in layer_pattern['positions']:
                loaded_items.append({
                    'position': (pos[0], pos[1], z),
                    'dims': box_dims,
                    'weight': box_weight
                })

        cog = check_center_of_gravity(loaded_items, self.pallet_dims)

        self.solution = {
            'pattern': pattern,
            'total_boxes': pattern['total_boxes'],
            'total_weight': total_weight,
            'total_height': pattern['num_layers'] * box_dims[2],
            'num_layers': pattern['num_layers'],
            'utilization': self.calculate_utilization(pattern, box_dims),
            'stability': cog,
            'loaded_items': loaded_items
        }

        return self.solution

    def optimize_mixed_sku(self):
        """
        Optimize loading for mixed SKUs

        Uses layer-by-layer heuristic
        """

        if len(self.items) < 2:
            raise ValueError("Use optimize_mixed_sku for multiple SKUs")

        # Sort items by volume (largest first)
        sorted_items = sorted(self.items,
                             key=lambda x: x['dims'][0] * x['dims'][1] * x['dims'][2],
                             reverse=True)

        loaded_items = []
        current_height = 0
        total_weight = 0

        # Load layer by layer
        while current_height < self.pallet_dims[2]:
            # Select item for this layer
            layer_item = None
            for item in sorted_items:
                if item['quantity'] > 0:
                    # Check if item fits
                    if current_height + item['dims'][2] <= self.pallet_dims[2]:
                        layer_item = item
                        break

            if layer_item is None:
                break  # No more items fit

            # Generate pattern for this layer
            layer_pattern = generate_block_pattern(
                layer_item['dims'],
                self.pallet_dims
            )

            boxes_to_load = min(
                layer_pattern['boxes_per_layer'],
                layer_item['quantity']
            )

            # Check weight
            if total_weight + boxes_to_load * layer_item['weight'] > self.weight_capacity:
                # Reduce boxes to fit weight
                boxes_to_load = int((self.weight_capacity - total_weight) / layer_item['weight'])

            if boxes_to_load == 0:
                break

            # Load boxes
            for i, pos in enumerate(layer_pattern['positions'][:boxes_to_load]):
                loaded_items.append({
                    'item_id': layer_item['id'],
                    'position': (pos[0], pos[1], current_height),
                    'dims': layer_item['dims'],
                    'weight': layer_item['weight']
                })

            layer_item['quantity'] -= boxes_to_load
            total_weight += boxes_to_load * layer_item['weight']
            current_height += layer_item['dims'][2]

        # Calculate metrics
        cog = check_center_of_gravity(loaded_items, self.pallet_dims)

        self.solution = {
            'total_boxes': len(loaded_items),
            'total_weight': total_weight,
            'total_height': current_height,
            'loaded_items': loaded_items,
            'stability': cog
        }

        return self.solution

    def calculate_utilization(self, pattern, box_dims):
        """Calculate volume utilization percentage"""
        box_volume = box_dims[0] * box_dims[1] * box_dims[2]
        total_box_volume = pattern['total_boxes'] * box_volume

        pallet_volume = (self.pallet_dims[0] * self.pallet_dims[1] *
                        pattern['num_layers'] * box_dims[2])

        return (total_box_volume / pallet_volume * 100) if pallet_volume > 0 else 0

    def visualize_3d(self, save_path=None):
        """3D visualization of pallet load"""

        if self.solution is None:
            raise ValueError("No solution to visualize. Run optimize first.")

        fig = plt.figure(figsize=(12, 10))
        ax = fig.add_subplot(111, projection='3d')

        # Draw pallet base
        pallet_l, pallet_w, pallet_h = self.pallet_dims
        self._draw_pallet_base(ax, pallet_l, pallet_w)

        # Draw loaded items
        colors = plt.cm.tab20(np.linspace(0, 1, 20))

        for idx, item in enumerate(self.solution['loaded_items']):
            pos = item['position']
            dims = item['dims']
            color = colors[idx % 20]

            self._draw_box_3d(ax, pos, dims, color, alpha=0.7)

        # Draw COG
        if self.solution.get('stability'):
            cog = self.solution['stability']['cog']
            ax.scatter([cog[0]], [cog[1]], [cog[2]],
                      c='red', s=200, marker='*',
                      label='Center of Gravity')

        ax.set_xlabel('Length (in)')
        ax.set_ylabel('Width (in)')
        ax.set_zlabel('Height (in)')
        ax.set_title(f'Pallet Load: {self.solution["total_boxes"]} boxes\n'
                    f'Weight: {self.solution["total_weight"]:.0f} lbs | '
                    f'Height: {self.solution["total_height"]:.1f} in')

        ax.legend()

        # Set equal aspect
        max_dim = max(pallet_l, pallet_w, self.solution['total_height'])
        ax.set_xlim(-5, max_dim + 5)
        ax.set_ylim(-5, max_dim + 5)
        ax.set_zlim(0, max_dim + 5)

        if save_path:
            plt.savefig(save_path, dpi=300, bbox_inches='tight')

        plt.show()

    def _draw_pallet_base(self, ax, length, width):
        """Draw pallet base"""
        vertices = np.array([
            [0, 0, 0], [length, 0, 0], [length, width, 0], [0, width, 0]
        ])

        face = [vertices]
        pallet_face = Poly3DCollection(face, facecolors='brown',
                                       linewidths=2, edgecolors='black',
                                       alpha=0.3)
        ax.add_collection3d(pallet_face)

    def _draw_box_3d(self, ax, position, dimensions, color, alpha=0.7):
        """Draw 3D box"""
        x, y, z = position
        l, w, h = dimensions

        vertices = np.array([
            [x, y, z], [x+l, y, z], [x+l, y+w, z], [x, y+w, z],
            [x, y, z+h], [x+l, y, z+h], [x+l, y+w, z+h], [x, y+w, z+h]
        ])

        faces = [
            [vertices[0], vertices[1], vertices[5], vertices[4]],
            [vertices[2], vertices[3], vertices[7], vertices[6]],
            [vertices[0], vertices[3], vertices[7], vertices[4]],
            [vertices[1], vertices[2], vertices[6], vertices[5]],
            [vertices[0], vertices[1], vertices[2], vertices[3]],
            [vertices[4], vertices[5], vertices[6], vertices[7]]
        ]

        face_collection = Poly3DCollection(faces, facecolors=color,
                                          linewidths=0.5, edgecolors='black',
                                          alpha=alpha)
        ax.add_collection3d(face_collection)

    def visualize_top_view(self, layer_index=0, save_path=None):
        """2D top-down view of specific layer"""

        if self.solution is None:
            raise ValueError("No solution to visualize")

        fig, ax = plt.subplots(figsize=(10, 8))

        # Draw pallet outline
        pallet_l, pallet_w = self.pallet_dims[:2]
        pallet_rect = patches.Rectangle((0, 0), pallet_l, pallet_w,
                                        linewidth=2, edgecolor='black',
                                        facecolor='lightgray', alpha=0.3)
        ax.add_patch(pallet_rect)

        # Filter items for this layer
        if 'pattern' in self.solution:
            # Single SKU
            if layer_index < len(self.solution['pattern']['layers']):
                layer = self.solution['pattern']['layers'][layer_index]
                box_dims = self.items[0]['dims']

                for pos in layer['positions']:
                    rect = patches.Rectangle(
                        (pos[0], pos[1]), pos[2], pos[3],
                        linewidth=1, edgecolor='blue',
                        facecolor='lightblue', alpha=0.7
                    )
                    ax.add_patch(rect)

                    # Add dimensions label
                    cx = pos[0] + pos[2]/2
                    cy = pos[1] + pos[3]/2
                    ax.text(cx, cy, f'{pos[2]:.0f}x{pos[3]:.0f}',
                           ha='center', va='center', fontsize=8)

        ax.set_xlim(-5, pallet_l + 5)
        ax.set_ylim(-5, pallet_w + 5)
        ax.set_aspect('equal')
        ax.set_xlabel('Length (in)')
        ax.set_ylabel('Width (in)')
        ax.set_title(f'Pallet Top View - Layer {layer_index + 1}')
        ax.grid(True, alpha=0.3)

        if save_path:
            plt.savefig(save_path, dpi=300, bbox_inches='tight')

        plt.show()

    def print_solution(self):
        """Print solution summary"""

        if self.solution is None:
            print("No solution available")
            return

        print("=" * 70)
        print("PALLET LOADING SOLUTION")
        print("=" * 70)
        print(f"Pallet type: {self.pallet_type}")
        print(f"Pallet dimensions: {self.pallet_dims[0]} x {self.pallet_dims[1]} x {self.pallet_dims[2]} in")
        print(f"Weight capacity: {self.weight_capacity} lbs")
        print()
        print(f"Total boxes loaded: {self.solution['total_boxes']}")
        print(f"Total weight: {self.solution['total_weight']:.1f} lbs "
              f"({self.solution['total_weight']/self.weight_capacity*100:.1f}% of capacity)")
        print(f"Total height: {self.solution['total_height']:.1f} in "
              f"({self.solution['total_height']/self.pallet_dims[2]*100:.1f}% of max)")

        if 'utilization' in self.solution:
            print(f"Volume utilization: {self.solution['utilization']:.1f}%")

        if self.solution.get('stability'):
            stab = self.solution['stability']
            print()
            print("Stability Analysis:")
            print(f"  Center of Gravity: ({stab['cog'][0]:.1f}, {stab['cog'][1]:.1f}, {stab['cog'][2]:.1f})")
            print(f"  Offset from center: ({stab['offset'][0]:.1f}, {stab['offset'][1]:.1f}) in")
            print(f"  Stability Score: {stab['stability_score']:.1f}/100")
            print(f"  Status: {'STABLE' if stab['is_stable'] else 'UNSTABLE - REVIEW REQUIRED'}")

# Example usage
if __name__ == "__main__":
    # Example 1: Single SKU optimization
    print("Example 1: Single SKU Pallet Loading")
    print("-" * 50)

    pallet = PalletOptimizer(pallet_type='GMA', weight_capacity=2500, max_height=72)

    # Add boxes (12x10x8 inches, 25 lbs each, 100 available)
    pallet.add_item(length=12, width=10, height=8, weight=25,
                   quantity=100, item_id='BOX_A',
                   compression_strength=250)

    # Optimize
    solution = pallet.optimize_single_sku(interlock=True)

    # Print results
    pallet.print_solution()

    # Visualize
    print("\nGenerating 3D visualization...")
    pallet.visualize_3d()

    print("\nGenerating top view...")
    pallet.visualize_top_view(layer_index=0)

Tools & Libraries

Python Libraries

palletizing - Pallet pattern generation py3dbp - Can be used for pallet loading rectpack - Rectangle packing (for layer generation)

Commercial Software

• CAPE PACK: Professional palletization software
• TOPS Pro: Pallet optimization
• PalletStacking: Online pallet calculator
• LoadCalc: Load planning software
• Pallet Design System (PDS): NWPCA software

Online Tools

• PackEx: Free pallet calculator
• PalletConfigr: Pallet pattern generator
• Boxologic: Free 3D packing tool

Common Challenges & Solutions

Challenge: Poor Pallet Utilization (<75%)

Problem:

• Boxes don't fit efficiently
• Large gaps between boxes
• Wasted vertical space

Solutions:

• Try different orientations (rotate 90°)
• Use interlocking pattern
• Mix similar-sized SKUs on same pallet
• Consider custom case sizes that tile better
• Fill gaps with smaller items

Challenge: Load Instability

Problem:

• Pallet tips during transport
• Boxes shift or collapse
• Heavy overhang

Solutions:

• Limit overhang to <3 inches
• Use interlocking pattern
• Place heavy items at bottom center
• Check center of gravity calculation
• Use slip sheets between layers
• Apply stretch wrap properly
• Add corner boards for support

Challenge: Compression Damage

Problem:

• Bottom boxes crushed
• Stack collapses during storage
• Corrugated boxes weakened by humidity

Solutions:

• Check compression strength specifications
• Limit stack height based on ECT rating
• Use column stacking for weak boxes
• Add load capping (top board)
• Control humidity in storage
• Use stronger boxes for bottom layers

Challenge: Mixed SKU Complexity

Problem:

• Different box sizes don't pack well together
• Height differences create instability
• Difficult to generate efficient patterns

Solutions:

• Group similar heights on same layer
• Use layer-by-layer approach
• Place largest/heaviest at bottom
• Fill small gaps with dunnage
• Consider using two pallets for very different sizes

Challenge: Shipping Constraints

Problem:

• Pallet too tall for truck
• Too heavy for dock equipment
• Doesn't fit through doors

Solutions:

• Set max height constraint (typically 72" or 84")
• Respect weight limits (forklift capacity 3,000-5,000 lbs)
• Check door clearances before optimization
• Consider double-stacking capability
• Plan for specific truck/rail car dimensions

Output Format

Pallet Loading Report

Summary:

• Pallet Type: GMA (48" x 40")
• Boxes Loaded: 84 cases
• Total Weight: 2,100 lbs
• Total Height: 64 inches
• Utilization: 89.5%
• Stability: PASS (COG offset: 1.2")

Loading Pattern:

Layer 1 (Bottom):

• Pattern: 4 x 5 = 20 boxes
• Orientation: 12"L x 10"W
• Layer Height: 8"
• Layer Weight: 500 lbs

Layer 2:

• Pattern: 5 x 4 = 20 boxes (rotated 90°)
• Orientation: 10"L x 12"W
• Layer Height: 16"
• Layer Weight: 500 lbs

Layer 3:

• Pattern: 4 x 5 = 20 boxes
• Orientation: 12"L x 10"W
• Layer Height: 24"
• Layer Weight: 500 lbs

Layer 4:

• Pattern: 5 x 4 = 20 boxes (rotated 90°)
• Orientation: 10"L x 12"W
• Layer Height: 32"
• Layer Weight: 500 lbs

Layer 5 (Top):

• Pattern: 2 x 2 = 4 boxes
• Orientation: 12"L x 10"W
• Layer Height: 40"
• Layer Weight: 100 lbs

Stability Analysis:

• Center of Gravity: (24.0", 20.0", 20.5")
• Offset from center: 1.2" horizontal
• Status: STABLE ✓
• Recommended: Stretch wrap with 3-4 passes

Shipping Instructions:

• Apply stretch wrap immediately after loading
• Use 4-way entry for forklift
• Can be double-stacked (weight allows)
• Max stack height: 2 high (128" total)
• Label all 4 sides

Questions to Ask

If you need more context:

1. What pallet size are you using? (GMA 48x40, Euro, custom?)
2. What are the box dimensions and weights?
3. How many boxes need to be loaded per pallet?
4. Is there a maximum height restriction?
5. Can boxes be rotated or mixed on same pallet?
6. Are there stability or compression strength concerns?
7. What's the shipping method? (truck, rail, container)
8. Do you need interlocking pattern for stability?

Related Skills

• 3d-bin-packing: For general 3D packing problems
• container-loading-optimization: For loading pallets into containers
• vehicle-loading-optimization: For truck loading with pallets
• 2d-bin-packing: For layer pattern generation
• load-building-optimization: For order fulfillment and pallet building
• warehouse-slotting-optimization: For pallet storage optimization