Problem: Minimum Total Waste Container Set

A pharmaceutical company needs to prepare liquid medication for multiple patient orders. Each order specifies a required volume. The company must select a single "container set" to fulfill all outstanding orders. A container set consists of a collection of available container capacities.

Rules for Fulfilling an Order

For each patient order with a required volume r, using the chosen container set, the following rules apply:

• Exactly one container must be chosen from the set.
• The chosen container must be filled completely.
• If a container with capacity exactly equal to r exists in the set, it must be used, resulting in zero waste for that order.
• Otherwise, the smallest available container capacity c from the set that is greater than or equal to r must be used.
• The waste generated for this specific order is calculated as c - r.
• If, for any order, a chosen container set does not contain a container capacity c such that c >= r, then that container set is considered invalid and cannot be selected.

Task

Given an array requirements representing the volumes needed for each patient order, and a list containerSets where containerSets[i] is an array of container capacities available in set i:

1. Compute the total waste for each valid container set (the sum of wastes across all orders).
2. Return the 0-based index of the container set that results in the minimum total waste.
3. In case of a tie for minimum total waste, return the smallest index among the tied sets.
4. If no container set can fulfill all orders (i.e., all sets are invalid), return -1.

Input/Output

• Input:
  • requirements: An array of positive integers representing required volumes.
  • containerSets: A list of arrays, where each inner array contains positive integers representing container capacities.
• Output: An integer, the 0-based index of the optimal container set, or -1.

Notes and Assumptions

• Container capacities and required volumes are positive integers.
• A container set may contain capacities in any order, and duplicate capacities are possible.
• The total number of capacities across all sets can be large (e.g., up to 2e5), implying that an efficient algorithmic approach is expected.