2D bin packing is a type of optimization problem that involves fitting a set of 2D rectangles (also known as “items”) into a single, larger rectangle (also known as the “bin”) in such a way that the items are arranged as efficiently as possible. It is used in a variety of applications, including:

• Packaging: 2D bin packing can be used to determine the most efficient way to pack a set of items into a container for shipping or storage.
• Scheduling: In manufacturing and other industries, 2D bin packing can be used to schedule the use of resources such as machine time or floor space.
• Data compression: 2D bin packing can be used to compress data by finding an efficient way to arrange the data in a fixed-size block.
• Layout design: 2D bin packing can be used to design efficient layouts for warehouses, factories, and other types of buildings.
• Graph drawing: In graph theory, 2D bin packing can be used to layout the nodes and edges of a graph in a way that minimizes the amount of space required.

An example of a 2D Bin Packing problem that tries to fit 5 rectangles (10x15, 20x25, 30x35, 40x45, 50x55) inside a larger rectangle (bin size of 100x200).

from ortools.linear_solver import pywraplp

def main():
  # Set the size of the bin and the sizes of the items.
  bin_width = 100
  bin_height = 200
  width = [10, 20, 30, 40, 50]
  height = [15, 25, 35, 45, 55]
  n = len(width)

  # Create the solver.
  solver = pywraplp.Solver('2D Bin Packing', pywraplp.Solver.GLOP_LINEAR_PROGRAMMING)

  # Create the variables.
  x = {}
  y = {}
  for i in range(n):
    x[i] = solver.NumVar(0, solver.infinity(), 'x[%i]' % i)
    y[i] = solver.NumVar(0, solver.infinity(), 'y[%i]' % i)

  # Set the objective function.
  objective = solver.Minimize(max(x) + max(y))

  # Set the constraints.
  for i in range(n):
    solver.Add(x[i] + width[i] <= bin_width)
    solver.Add(y[i] + height[i] <= bin_height)

  # Solve the problem.
  status = solver.Solve()

  # Print the solution.
  if status == pywraplp.Solver.OPTIMAL:
    print('Optimal solution found.')
    for i in range(n):
      print('Item', i, ':', 'x =', x[i].solution_value(), 'y =', y[i].solution_value())
  else:
    print('Problem could not be solved.')

if __name__ == '__main__':
  main()