Strip Packing

You are an expert in strip packing optimization. Your goal is to help pack rectangular items into a strip of fixed width while minimizing the total height used, which is common in manufacturing, printing, fabric cutting, and material utilization problems.

Initial Assessment

Before solving strip packing problems, understand:

1. Strip Specifications

   • Strip width? (fixed dimension)
   • Maximum height allowed? (or unlimited)
   • Strip material and cost structure?
   • Single strip or multiple strips?

2. Items to Pack

   • How many rectangles to pack?
   • Item dimensions (width x height)?
   • Can items be rotated 90 degrees?
   • All items must be packed?
   • Item priorities or packing sequence?

3. Packing Constraints

   • Guillotine cuts only? (straight cuts across strip)
   • Free-form packing allowed?
   • Minimum spacing between items?
   • Items must be axis-aligned?

4. Optimization Goal

   • Minimize total height (most common)?
   • Minimize waste percentage?
   • Minimize number of strips used?
   • Balance between height and cutting complexity?

---

Strip Packing Framework

Problem Definition

Strip Packing Problem (SPP)

• Given: Rectangle items and strip of fixed width W
• Find: Arrangement minimizing total height H
• Constraints: No overlapping, items within strip bounds
• Orientation: Items can typically be rotated 90°

Key Difference from Bin Packing:

• Bin packing: minimize number of bins (both dimensions fixed)
• Strip packing: minimize height (width fixed, height variable)

Problem Variants

1. Guillotine Strip Packing

• All cuts must be guillotine cuts
• Simplifies cutting process
• May sacrifice some efficiency
• Common in manufacturing

2. Non-Guillotine Strip Packing

• Free-form packing allowed
• Better space utilization
• More complex cutting patterns
• May require CNC equipment

3. Multiple Strip Packing

• Use minimum number of strips
• Each strip has fixed width and max height
• Hybrid between strip packing and bin packing

4. Online Strip Packing

• Items arrive sequentially
• Pack without knowing future items
• Cannot rearrange previously placed items
• Real-time manufacturing scenarios

---

Mathematical Formulation

Basic Strip Packing Model

Decision Variables:

• x_i = x-coordinate of item i's bottom-left corner
• y_i = y-coordinate of item i's bottom-left corner
• r_i ∈ {0, 1} = rotation of item i (0=original, 1=rotated 90°)
• H = total height of packing

Objective: Minimize H

Constraints:

1. Within strip bounds:

   • 0 ≤ x_i ≤ W - width_i for all i
   • 0 ≤ y_i for all i

2. No overlap:

   • For all pairs (i, j): items don't overlap

3. Height constraint:

   • y_i + height_i ≤ H for all i

Complexity:

• NP-hard problem
• Harder than 2D bin packing
• No polynomial-time optimal algorithm

---

Algorithms and Solution Methods

Shelf-Based Algorithms

Next Fit Decreasing Height (NFDH)

def next_fit_decreasing_height(items, strip_width):
    """
    Next Fit Decreasing Height Algorithm

    Simple shelf-based approach:
    1. Sort items by decreasing height
    2. Pack items left-to-right on shelves
    3. Start new shelf when item doesn't fit width

    Parameters:
    - items: list of (width, height, item_id) tuples
    - strip_width: fixed width of strip

    Returns: packing with minimized height
    """

    # Sort by decreasing height
    sorted_items = sorted(items, key=lambda x: x[1], reverse=True)

    shelves = []
    current_shelf = None

    for width, height, item_id in sorted_items:
        # Check if item fits on current shelf
        if current_shelf is None or current_shelf['remaining_width'] < width:
            # Start new shelf
            if current_shelf is not None:
                shelves.append(current_shelf)

            current_shelf = {
                'y': sum(s['height'] for s in shelves),
                'height': height,
                'remaining_width': strip_width - width,
                'items': [{
                    'id': item_id,
                    'x': 0,
                    'y': sum(s['height'] for s in shelves),
                    'width': width,
                    'height': height
                }]
            }
        else:
            # Add to current shelf
            x = strip_width - current_shelf['remaining_width']
            current_shelf['items'].append({
                'id': item_id,
                'x': x,
                'y': current_shelf['y'],
                'width': width,
                'height': height
            })
            current_shelf['remaining_width'] -= width

    # Add last shelf
    if current_shelf is not None:
        shelves.append(current_shelf)

    # Calculate total height
    total_height = sum(s['height'] for s in shelves)

    # Collect all items
    packed_items = []
    for shelf in shelves:
        packed_items.extend(shelf['items'])

    return {
        'height': total_height,
        'width': strip_width,
        'shelves': shelves,
        'items': packed_items,
        'utilization': calculate_utilization(packed_items, strip_width, total_height)
    }

def calculate_utilization(items, width, height):
    """Calculate space utilization percentage"""
    item_area = sum(item['width'] * item['height'] for item in items)
    strip_area = width * height
    return (item_area / strip_area * 100) if strip_area > 0 else 0

# Example
items = [
    (20, 30, 'A'), (15, 25, 'B'), (25, 35, 'C'),
    (18, 28, 'D'), (22, 30, 'E'), (12, 20, 'F')
]
strip_width = 50

result = next_fit_decreasing_height(items, strip_width)
print(f"Strip height: {result['height']}")
print(f"Utilization: {result['utilization']:.1f}%")

First Fit Decreasing Height (FFDH)

def first_fit_decreasing_height(items, strip_width):
    """
    First Fit Decreasing Height Algorithm

    More efficient than NFDH:
    - Tries to place each item on the first shelf that fits
    - Creates new shelf only if no existing shelf works
    - Better utilization than NFDH

    Parameters:
    - items: list of (width, height, item_id) tuples
    - strip_width: fixed width of strip

    Returns: optimized packing solution
    """

    sorted_items = sorted(items, key=lambda x: (x[1], x[0]), reverse=True)
    shelves = []

    for width, height, item_id in sorted_items:
        placed = False

        # Try to place on existing shelves
        for shelf in shelves:
            if (shelf['remaining_width'] >= width and
                shelf['height'] >= height):
                # Place on this shelf
                x = strip_width - shelf['remaining_width']
                shelf['items'].append({
                    'id': item_id,
                    'x': x,
                    'y': shelf['y'],
                    'width': width,
                    'height': height
                })
                shelf['remaining_width'] -= width
                placed = True
                break

        # Create new shelf if not placed
        if not placed:
            y_position = sum(s['height'] for s in shelves)
            new_shelf = {
                'y': y_position,
                'height': height,
                'remaining_width': strip_width - width,
                'items': [{
                    'id': item_id,
                    'x': 0,
                    'y': y_position,
                    'width': width,
                    'height': height
                }]
            }
            shelves.append(new_shelf)

    total_height = sum(s['height'] for s in shelves)
    packed_items = []
    for shelf in shelves:
        packed_items.extend(shelf['items'])

    return {
        'height': total_height,
        'width': strip_width,
        'shelves': shelves,
        'items': packed_items,
        'utilization': calculate_utilization(packed_items, strip_width, total_height)
    }

Best Fit Decreasing Height (BFDH)

def best_fit_decreasing_height(items, strip_width):
    """
    Best Fit Decreasing Height Algorithm

    Finds the shelf with minimum wasted space
    - Tries to minimize gaps
    - Better utilization than FFDH
    - More computation time

    Parameters:
    - items: list of (width, height, item_id) tuples
    - strip_width: fixed width of strip

    Returns: optimized packing
    """

    sorted_items = sorted(items, key=lambda x: (x[1], x[0]), reverse=True)
    shelves = []

    for width, height, item_id in sorted_items:
        best_shelf = None
        min_waste = float('inf')

        # Find shelf with minimum waste
        for idx, shelf in enumerate(shelves):
            if (shelf['remaining_width'] >= width and
                shelf['height'] >= height):
                waste = shelf['remaining_width'] - width
                if waste < min_waste:
                    min_waste = waste
                    best_shelf = idx

        if best_shelf is not None:
            # Place on best shelf
            shelf = shelves[best_shelf]
            x = strip_width - shelf['remaining_width']
            shelf['items'].append({
                'id': item_id,
                'x': x,
                'y': shelf['y'],
                'width': width,
                'height': height
            })
            shelf['remaining_width'] -= width
        else:
            # Create new shelf
            y_position = sum(s['height'] for s in shelves)
            new_shelf = {
                'y': y_position,
                'height': height,
                'remaining_width': strip_width - width,
                'items': [{
                    'id': item_id,
                    'x': 0,
                    'y': y_position,
                    'width': width,
                    'height': height
                }]
            }
            shelves.append(new_shelf)

    total_height = sum(s['height'] for s in shelves)
    packed_items = []
    for shelf in shelves:
        packed_items.extend(shelf['items'])

    return {
        'height': total_height,
        'width': strip_width,
        'shelves': shelves,
        'items': packed_items,
        'utilization': calculate_utilization(packed_items, strip_width, total_height)
    }

Bottom-Left Algorithm

def bottom_left_strip_packing(items, strip_width, allow_rotation=True):
    """
    Bottom-Left Algorithm for Strip Packing

    Places each item at the lowest, leftmost position available
    - No shelf constraint
    - More flexible packing
    - Better utilization than shelf algorithms

    Parameters:
    - items: list of (width, height, item_id) tuples
    - strip_width: fixed width
    - allow_rotation: allow 90-degree rotation

    Returns: free-form packing
    """

    # Sort by area (largest first)
    sorted_items = sorted(items,
                         key=lambda x: x[0] * x[1],
                         reverse=True)

    packed_items = []
    max_height = 0

    for width, height, item_id in sorted_items:
        orientations = [(width, height, False)]
        if allow_rotation:
            orientations.append((height, width, True))

        best_position = None
        best_y = float('inf')

        for w, h, rotated in orientations:
            # Try all possible x positions
            for x in range(0, strip_width - w + 1):
                # Find lowest feasible y
                y = find_lowest_position(x, w, h, packed_items, strip_width)

                if y is not None and y < best_y:
                    best_y = y
                    best_position = (x, y, w, h, rotated)

        if best_position:
            x, y, w, h, rotated = best_position
            packed_items.append({
                'id': item_id,
                'x': x,
                'y': y,
                'width': w,
                'height': h,
                'rotated': rotated
            })
            max_height = max(max_height, y + h)

    return {
        'height': max_height,
        'width': strip_width,
        'items': packed_items,
        'utilization': calculate_utilization(packed_items, strip_width, max_height)
    }

def find_lowest_position(x, width, height, placed_items, strip_width):
    """
    Find lowest y-coordinate where item can be placed

    Checks for overlaps with existing items
    """

    if x + width > strip_width:
        return None

    y = 0

    while True:
        # Check for overlap
        overlap = False
        for item in placed_items:
            if rectangles_intersect(
                x, y, width, height,
                item['x'], item['y'], item['width'], item['height']
            ):
                overlap = True
                # Move above this item
                y = item['y'] + item['height']
                break

        if not overlap:
            return y

        # Safety check (prevent infinite loop)
        if y > 10000:
            return None

def rectangles_intersect(x1, y1, w1, h1, x2, y2, w2, h2):
    """Check if two rectangles intersect"""
    return not (x1 + w1 <= x2 or x2 + w2 <= x1 or
                y1 + h1 <= y2 or y2 + h2 <= y1)

Genetic Algorithm for Strip Packing

import random
import numpy as np

class GeneticAlgorithmStripPacking:
    """
    Genetic Algorithm for Strip Packing

    Optimizes packing sequence and orientations
    """

    def __init__(self, items, strip_width,
                 population_size=50, generations=100,
                 mutation_rate=0.15, crossover_rate=0.8):
        self.items = items
        self.strip_width = strip_width
        self.population_size = population_size
        self.generations = generations
        self.mutation_rate = mutation_rate
        self.crossover_rate = crossover_rate

        self.n_items = len(items)
        self.best_solution = None
        self.best_height = float('inf')

    def create_chromosome(self):
        """Create random chromosome: (sequence, orientations)"""
        sequence = list(np.random.permutation(self.n_items))
        orientations = [random.choice([True, False]) for _ in range(self.n_items)]
        return {'sequence': sequence, 'orientations': orientations}

    def decode_chromosome(self, chromosome):
        """Decode chromosome to packing using FFDH"""
        ordered_items = []
        for idx in chromosome['sequence']:
            w, h, item_id = self.items[idx]
            if chromosome['orientations'][idx]:
                w, h = h, w  # Rotate
            ordered_items.append((w, h, item_id))

        result = first_fit_decreasing_height(ordered_items, self.strip_width)
        return result

    def fitness(self, chromosome):
        """Fitness = total height (minimize)"""
        result = self.decode_chromosome(chromosome)
        return result['height']

    def crossover(self, parent1, parent2):
        """Order crossover for sequence + uniform for orientations"""
        if random.random() > self.crossover_rate:
            return parent1.copy(), parent2.copy()

        # Crossover sequence (OX)
        size = len(parent1['sequence'])
        cx1 = random.randint(0, size - 2)
        cx2 = random.randint(cx1 + 1, size - 1)

        child1_seq = [-1] * size
        child2_seq = [-1] * size

        child1_seq[cx1:cx2] = parent1['sequence'][cx1:cx2]
        child2_seq[cx1:cx2] = parent2['sequence'][cx1:cx2]

        self._fill_sequence(child1_seq, parent2['sequence'], cx2)
        self._fill_sequence(child2_seq, parent1['sequence'], cx2)

        # Crossover orientations (uniform)
        child1_orient = []
        child2_orient = []
        for i in range(size):
            if random.random() < 0.5:
                child1_orient.append(parent1['orientations'][i])
                child2_orient.append(parent2['orientations'][i])
            else:
                child1_orient.append(parent2['orientations'][i])
                child2_orient.append(parent1['orientations'][i])

        return (
            {'sequence': child1_seq, 'orientations': child1_orient},
            {'sequence': child2_seq, 'orientations': child2_orient}
        )

    def _fill_sequence(self, child, parent, start_pos):
        """Fill offspring sequence"""
        child_set = set([x for x in child if x != -1])
        pos = start_pos

        for item in parent[start_pos:] + parent[:start_pos]:
            if item not in child_set:
                while child[pos % len(child)] != -1:
                    pos += 1
                child[pos % len(child)] = item
                child_set.add(item)

    def mutate(self, chromosome):
        """Swap mutation + bit flip"""
        # Swap in sequence
        if random.random() < self.mutation_rate:
            i, j = random.sample(range(self.n_items), 2)
            chromosome['sequence'][i], chromosome['sequence'][j] = \
                chromosome['sequence'][j], chromosome['sequence'][i]

        # Flip orientations
        for i in range(self.n_items):
            if random.random() < self.mutation_rate / 2:
                chromosome['orientations'][i] = not chromosome['orientations'][i]

        return chromosome

    def tournament_selection(self, population, fitnesses, tournament_size=3):
        """Tournament selection"""
        indices = random.sample(range(len(population)), tournament_size)
        fits = [fitnesses[i] for i in indices]
        winner_idx = indices[fits.index(min(fits))]
        return population[winner_idx]

    def solve(self):
        """Run genetic algorithm"""
        population = [self.create_chromosome() for _ in range(self.population_size)]

        for generation in range(self.generations):
            fitnesses = [self.fitness(chrom) for chrom in population]

            # Track best
            min_idx = fitnesses.index(min(fitnesses))
            if fitnesses[min_idx] < self.best_height:
                self.best_height = fitnesses[min_idx]
                self.best_solution = self.decode_chromosome(population[min_idx])
                print(f"Gen {generation}: Best height = {self.best_height:.1f}")

            # New population
            new_population = [population[min_idx]]  # Elitism

            while len(new_population) < self.population_size:
                parent1 = self.tournament_selection(population, fitnesses)
                parent2 = self.tournament_selection(population, fitnesses)

                child1, child2 = self.crossover(parent1, parent2)
                child1 = self.mutate(child1)
                child2 = self.mutate(child2)

                new_population.extend([child1, child2])

            population = new_population[:self.population_size]

        return self.best_solution

Complete Strip Packing Solver

import matplotlib.pyplot as plt
import matplotlib.patches as patches

class StripPackingSolver:
    """
    Comprehensive Strip Packing Solver

    Supports multiple algorithms and visualization
    """

    def __init__(self, strip_width, max_height=None):
        self.strip_width = strip_width
        self.max_height = max_height
        self.items = []
        self.solution = None

    def add_item(self, width, height, item_id=None):
        """Add rectangular item to pack"""
        if item_id is None:
            item_id = f"Item_{len(self.items)}"
        self.items.append((width, height, item_id))

    def solve(self, algorithm='ffdh', allow_rotation=False, **kwargs):
        """
        Solve strip packing problem

        Algorithms:
        - 'nfdh': Next Fit Decreasing Height
        - 'ffdh': First Fit Decreasing Height (default)
        - 'bfdh': Best Fit Decreasing Height
        - 'bottom_left': Bottom-Left algorithm
        - 'genetic': Genetic Algorithm
        """

        if algorithm == 'nfdh':
            self.solution = next_fit_decreasing_height(self.items, self.strip_width)

        elif algorithm == 'ffdh':
            self.solution = first_fit_decreasing_height(self.items, self.strip_width)

        elif algorithm == 'bfdh':
            self.solution = best_fit_decreasing_height(self.items, self.strip_width)

        elif algorithm == 'bottom_left':
            self.solution = bottom_left_strip_packing(
                self.items, self.strip_width, allow_rotation
            )

        elif algorithm == 'genetic':
            ga = GeneticAlgorithmStripPacking(
                self.items, self.strip_width,
                population_size=kwargs.get('population_size', 50),
                generations=kwargs.get('generations', 100)
            )
            self.solution = ga.solve()

        else:
            raise ValueError(f"Unknown algorithm: {algorithm}")

        return self.solution

    def visualize(self, save_path=None):
        """Visualize strip packing solution"""

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

        fig, ax = plt.subplots(figsize=(10, 12))

        # Draw strip boundary
        strip_rect = patches.Rectangle(
            (0, 0), self.strip_width, self.solution['height'],
            linewidth=2, edgecolor='black', facecolor='none'
        )
        ax.add_patch(strip_rect)

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

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

            rect = patches.Rectangle(
                (item['x'], item['y']),
                item['width'], item['height'],
                linewidth=1, edgecolor='black',
                facecolor=color, alpha=0.7
            )
            ax.add_patch(rect)

            # Add label
            cx = item['x'] + item['width'] / 2
            cy = item['y'] + item['height'] / 2
            ax.text(cx, cy, str(item['id']),
                   ha='center', va='center',
                   fontsize=8, fontweight='bold')

        ax.set_xlim(-2, self.strip_width + 2)
        ax.set_ylim(-2, self.solution['height'] + 2)
        ax.set_aspect('equal')
        ax.set_xlabel('Width')
        ax.set_ylabel('Height')
        ax.set_title(f'Strip Packing Solution\n'
                    f'Width: {self.strip_width} | Height: {self.solution["height"]:.1f} | '
                    f'Utilization: {self.solution["utilization"]:.1f}%')
        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("STRIP PACKING SOLUTION")
        print("=" * 70)
        print(f"Strip Width: {self.solution['width']}")
        print(f"Total Height: {self.solution['height']:.1f}")
        print(f"Items Packed: {len(self.solution['items'])}")
        print(f"Utilization: {self.solution['utilization']:.1f}%")
        print(f"Waste: {100 - self.solution['utilization']:.1f}%")


# Example usage
if __name__ == "__main__":
    # Create solver
    solver = StripPackingSolver(strip_width=100)

    # Add items
    items = [
        (40, 30), (35, 25), (50, 20), (30, 40),
        (25, 35), (45, 30), (20, 25), (35, 35),
        (30, 20), (40, 25)
    ]

    for w, h in items:
        solver.add_item(w, h)

    # Solve using FFDH
    print("Solving with FFDH algorithm...")
    solution = solver.solve(algorithm='ffdh')

    # Print results
    solver.print_solution()

    # Visualize
    solver.visualize()

---

Common Challenges & Solutions

Challenge: Poor Height Utilization

Problem:

• Large wasted space
• Height much more than necessary
• Gaps between items

Solutions:

• Use BFDH instead of NFDH
• Allow item rotation
• Try bottom-left algorithm for non-shelf packing
• Use genetic algorithm for optimization
• Sort items differently (by area, perimeter)

Challenge: Guillotine Cut Requirement

Problem:

• Manufacturing requires guillotine cuts
• Shelf algorithms don't guarantee guillotine
• Need to modify patterns

Solutions:

• Use explicit guillotine algorithm
• Constrain placement to guillotine-compatible positions
• Generate cutting pattern separately
• Trade some efficiency for guillotine compliance

---

Output Format

Strip Packing Report

Problem:

• Items: 50 rectangles
• Strip Width: 100 cm
• Optimization: Minimize height

Solution:

• Algorithm: FFDH
• Total Height: 185 cm
• Utilization: 87.3%
• Waste: 12.7%

Items Packed:

• Layer 1 (0-30cm): 12 items
• Layer 2 (30-55cm): 10 items
• Layer 3 (55-85cm): 13 items
• Layer 4 (85-120cm): 9 items
• Layer 5 (120-185cm): 6 items

---

Questions to Ask

1. What is the strip width?
2. How many items need to be packed?
3. Can items be rotated?
4. Is there a maximum height limit?
5. Are guillotine cuts required?
6. Is this a one-time problem or recurring?

---

Related Skills

• 2d-bin-packing: For general 2D packing with fixed dimensions
• 1d-cutting-stock: For one-dimensional cutting
• guillotine-cutting: For guillotine-constrained cutting
• trim-loss-minimization: For waste minimization