在线装箱与背包混合问题定义问询：固定箱数与固定物品数场景

Let me walk through this hybrid optimization problem you're unpacking—your constraints make this a unique mix of two classic problems, so let's break it down clearly:

Core Constraints Recap #

• Fixed number of bins: N total bins to fill
• Each bin must hold exactly k items (no more, no less)
• Objective: Minimize the total deviation between each bin's total volume and a target value V (whether it's absolute deviation, squared deviation, etc., will tweak the approach, but we'll stick to the core goal for now)
• Online problem characteristic: Items arrive sequentially; you can't look ahead to future items
• Unlimited item supply: Each item has a unique volume v_i, and you'll never run out of items to fill all N bins to k items each

Why This Is a Hybrid Problem #

This sits right at the intersection of online bin packing and knapsack:

• Online bin packing flavor: You have to make decisions in real-time as items arrive, with no foresight. Unlike traditional bin packing (where you minimize the number of bins), here you have a fixed bin count and strict per-bin item limits.
• Knapsack flavor: Each bin acts like a constrained knapsack—you need to select exactly k items to get as close as possible to the target volume V. The twist is you're solving this knapsack problem N times, in an online fashion, as items come in.

Initial Approach Ideas #

Since this is an online problem, greedy strategies are usually the starting point (optimal offline solutions won't apply here because you can't pre-plan all bins):

1. Greedy bin assignment: For each incoming item, assign it to the bin that currently has:
   • Fewer than k items already
   • A total volume that, when added to the item's v_i, brings it closest to V (minimizing immediate deviation)
   • If multiple bins fit this, pick the one with the least number of items so far to prioritize filling bins to k items quickly
2. Deviation weighting: If your objective prioritizes minimizing large deviations over small ones (e.g., squared deviation), adjust your greedy assignment to prioritize bins where adding the item would reduce the largest current deviation first.
3. Item reuse flexibility: Since items are unlimited, you don't have to stress about "wasting" a good item on a suboptimal bin—you'll get another chance to use it later for a better fit. This simplifies decision-making compared to finite-item scenarios.

内容的提问来源于stack exchange，提问作者MartinCunn24