Algorithm for Box Selection / Space Optimization

Question

So, I have an optimization/space management problem. This same question is posted on two SE sites simultaneously, Stack Overflow and Mathematics, since I think it is fitting for both.

Let's say I have containers of (X, Y) dimensions. Inside, I want to place as many smaller containers of (x_i, y_i) dimensions as possible. These smaller containers can vary in size. Inside these smaller containers, I have a given item which is the smallest unit I have.

Basically: Items, inside variable small boxes, inside variable big boxes.

Provided I know all of the following:

• Larger containers' dimensions.
• Smaller containers' dimensions.
• Amount of all containers available to me.
• Amount of basic items that need to be packaged.

Is there any known algorithm that can be applied to determine the following:

• The best selection of small boxes to be used (they can vary in size).
• The optimal number of large containers required for storage.
• The appropriate placement of the smaller packages inside the larger one.

IF this is a thing that already exists, can it be extended to three dimensions, rather than only two, as my example states? That is to say, calculate packaging configurations for dimensions (x, y, z).

EXAMPLE: I have

• 25 total items to pack
• 3 small containers with dimension: 3x3, 2x2, 1x2
• Dimension of the large container: 5x5

Through trial and error (or intuition?), I can see that a valid configuration is as follows:

• 1 small container of 3x3
• 3 small containers of 2x2
• 2 small containers of 1x2

All inside a single 5x5 container.

Now, it's a matter of tetrising/tangraming my way into a solution. So, for example, the following two are valid placements/configurations:

While this one isn't since it would require a second large container:

Answer 1

This is the well known rectangle packing problem or Korf's problem. Here is a simple solution with MiniZinc with the restriction of not turning the rectangles. Other answers here.