Question

An analytics firm processes data batches using two types of cloud computing instances: High-Memory Instances ($M$) and High-Compute Instances ($C$).

- Processing Capacity: Each High-Memory Instance processes $120$ thousand transactions per hour, and each High-Compute Instance processes $200$ thousand transactions per hour. The workload requires a total processing rate of at least $1,320$ thousand transactions per hour.
- Instance Availability: At most $10$ High-Memory Instances ($M \le 10$) and at most $6$ High-Compute Instances ($C \le 6$) are available. At least $1$ instance of each type must be used ($M \ge 1$ and $C \ge 1$).
- Load Balancing Constraint: To ensure infrastructure stability, the number of High-Memory Instances cannot exceed twice the number of High-Compute Instances ($M \le 2C$).
- Operating Costs: High-Memory Instances cost $\$18$ per hour each, while High-Compute Instances cost $\$25$ per hour each.

Select the number of High-Memory Instances ($M$) and the number of High-Compute Instances ($C$) that minimize the total hourly operating cost while satisfying all operational requirements.