• There are 100 items to be sorted into just over 100 bins, where no bin has more than one item and not every bin is necessarily filled, but every item must be placed.
• There are certain items that are of high enough "quality" to be placed in any bin, and there are also items that may not be of high enough quality for certain high-quality exclusive bins. There are no low-quality exclusive bins.
• Bins are only open for certain intervals throughout the day, and thus certain items are not compatible with certain bins because the items are not available during the open-bin times. When a bin is not open, the item may not be placed inside.
• Items also have preferences over which bins they are placed in, and likewise bins have preferences over which items they are to hold.
Assuming that all of the required information about each "item" and "bin" is known, what is the best method of sorting large numbers of items into roughly the same number of bins so that every item has a bin, there are no placements that make an item incompatible with a bin (and vice versa), and to maximize the total "happiness" in each item-bin pair?