I have a situation where I would like your input and think over / solve algorithm.

The specs

There are inventories with a specified amount of capacity.
‘X
(width) and Y (time)’ creates together a total space. ’60 * 100 =
6000′. In reality “width” are a business value unit, but we can talk width here.

Now there also are boxes, which take a part of the space.
Each
box take width (X) and time (Y). Two boxes with 80 * 20 each, will
fill a inventory with 1600+1600. It will result a fill factor, 2800
space left in the inventory, after the boxes are added. Each box
always belong to it’s inventory.

There exist unlimited, serial dependent, inventories with boxes added individually.
The former inventory have to be performed before next are performed.

Boxes themself have a attribute, like a color, which make a ownership
in higher context. Like a “give me all boxes of color -yellow-”
context. I.e. to find out which inventories contain that color. A box
with same color can exist several times, even in same inventory. But
if in same inventory, they have unique box-id’s.

The problem

There have to be added optimization strategy which make width (X) adjustable. Time (Y) is still fixed and can’t change. The optimization
rule is that available space in former inventory should be filled with
a part/slice from second inventory.

Time must fit. I.e. 2×15 or 1×25 can fit to 1×30. 1×35 can’t fit. Also the width added to former, must be reduced from second.
Remember that a box isn’t allowed to be removed from it’s inventory (but width can be
1 as minimum). Though, on adjustment, new boxes have to be created in the former
inventory.

It’s a mathematical theory but i mainly want to transform a algorithm of this kind, into code. Or give the question a glance of what kind of theories to look for. Perhaps this is a common problem? or too complex considered as “re-invent the wheel” and recommend to buy a third party module.

[Edit] ‘length’ changed to ‘width’
Added a graphic to better illustrate how the optimization rule will apply.

There are a little green piece remaind at right, after optimization, because of the “at least 1 width must remain”-rule. The red and blue did not fit fully.