You are an operations research specialist scheduling an aero...

Prompt

You are an operations research specialist scheduling an aerospace assembly hangar. Solve the following deterministic scheduling problem to find the absolute minimum total time (makespan) to complete all jobs. Workstations: - Station 1: Precision Milling (M1) - Station 2: Thermal Coating (M2) Rules: 1. Each station can process at most one component at a time. 2. Operations cannot be interrupted or preempted once started. 3. Station 1 requires NO cleaning/changeover time between components. 4. Station 2 requires exactly 1 hour of cleaning whenever switching from one component to a different component. 5. All components are available at Hour 0. Each component must follow its exact operational sequence in strict order (operation 2 cannot start until operation 1 finishes, etc.). Components to manufacture: - Alpha: Step 1: Station 1 for 4 hours Step 2: Station 2 for 3 hours - Beta: Step 1: Station 2 for 5 hours Step 2: Station 1 for 2 hours - Gamma: Step 1: Station 1 for 3 hours Step 2: Station 2 for 4 hours Step 3: Station 1 for 1 hour - Delta: Step 1: Station 2 for 2 hours Step 2: Station 1 for 4 hours Tasks: 1. State the absolute minimum total makespan (in hours) to finish all 4 components. 2. Provide the complete hour-by-hour timeline for Station 1 and Station 2. 3. Provide a brief mathematical proof demonstrating why this makespan is the theoretical lower bound.