Add tasks: (1)bin-packing-optimum-bins (2) gap-min-cost-assignment (3) lot-sizing-optimum-cost

[JianchengZ]

(1)bin-packing-optimum-bins

What it’s about: This task asks the agent to determine the minimum number of bins needed to pack a specific list of ten items with fixed sizes into bins of capacity 10. It is a classic one‑dimensional bin‑packing problem with a unique optimal answer.
Skills tested: The agent must perform combinatorial reasoning, apply lower‑bound estimates, and search or construct feasible packings. It tests the ability to reason about subset combinations and to verify that a candidate packing meets capacity and minimality constraints.
Interesting challenges: The main challenge is ensuring the agent finds the optimal solution (6 bins) rather than stopping at a heuristic solution. Because the search space grows exponentially, the task is designed so that naive enumeration is still feasible but requires careful reasoning or pruning.

(2) gap-min-cost-assignment

What it’s about: Here the agent must assign five jobs to three agents, each with its own capacity, resource requirements, and cost matrix, so as to minimise total cost without exceeding any agent’s capacity. It is a small instance of the Generalised Assignment Problem (GAP).
Skills tested: The task exercises discrete optimisation skills—specifically the ability to evaluate cost/resource trade‑offs, respect capacity constraints, and select an optimal assignment from a finite but non‑trivial search space.
Interesting challenges: The optimal assignment requires balancing low costs against the agents’ limited capacities and may not be obvious from the raw data. A brute‑force search is possible but care must be taken to prune infeasible assignments and verify the cost calculations correctly.

(3) lot-sizing-optimum-cost

What it’s about: This task involves planning production for a single item across six periods with varying demand, capacity, variable costs, setup costs, and a holding cost. The agent must choose production quantities to meet demand exactly, start and end with zero inventory, and minimise total cost.
Skills tested: It tests dynamic planning and optimisation abilities—specifically, understanding inventory balance constraints, fixed versus variable production costs, and the interplay between producing early (incurring holding costs) versus delaying production (subject to capacity limits and setup costs).
Interesting challenges: The optimal solution requires reasoning across multiple periods, deciding when it is economical to produce extra units and carry inventory, and ensuring end‑of‑horizon conditions are met. Designing the instance so that the optimal cost is non‑trivial yet deterministic required careful tuning of demands and capacities.