Size Match: Match Items to Containers Using Bin-Packing Heuristics

Problem You have n items with sizes and k bins each with a fixed capacity. Assign each item to exactly one bin such that no bin exceeds its capacity, and the number of bins used is minimized. Items cannot be split. Example Follow-ups First-Fit Decreasing gives at most (11/9)OPT + 6/9 bins — can you prove why? How do you solve this exactly using dynamic programming or ILP for small inputs? How does the problem change if bins have both weight and volume constraints?