Parts P1-P7 are machined first on a milling machine and then polished at a separate machine. Using the information in the following table, the minimum total completion time required for carrying out both the operations for all 7 parts is __________ hours.

Part	Milling (hours)	Polishing (hours)
P1	8	6
P2	3	2
P3	3	4
P4	4	6
P5	5	7
P6	6	4
P7	2	1

The correct option is 33.

To find the minimum total completion time (makespan) for processing a set of jobs on two sequential machines (Milling followed by Polishing), we use Johnson's Rule.

Step 1: Identify the processing times for each part
The times (in hours) given in the table are:
• Part P1: Milling = 8, Polishing = 6
• Part P2: Milling = 3, Polishing = 2
• Part P3: Milling = 3, Polishing = 4
• Part P4: Milling = 4, Polishing = 6
• Part P5: Milling = 5, Polishing = 7
• Part P6: Milling = 6, Polishing = 4
• Part P7: Milling = 2, Polishing = 1

Step 2: Determine the optimal sequence using Johnson's Algorithm
We look for the minimum processing time among all remaining parts:
1. The minimum time overall is 1 hour for P7 on Polishing (Machine 2). Since it is on Machine 2, place P7 at the end of the sequence:
[_, _, _, _, _, _, P7]
2. The next minimum time is 2 hours for P2 on Polishing (Machine 2). Place P2 at the remaining last position:
[_, _, _, _, _, P2, P7]
3. The next minimum time is 3 hours. This occurs for P3 on Milling (Machine 1) and P2 (already scheduled). Placing P3 at the start of the sequence:
[P3, _, _, _, _, P2, P7]
4. The next minimum time is 4 hours, which occurs for P4 on Milling (Machine 1) and P6 on Polishing (Machine 2). Place P4 at the beginning of the remaining slots, and P6 at the end of the remaining slots:
[P3, P4, _, _, P6, P2, P7]
5. The next minimum time is 5 hours for P5 on Milling (Machine 1). Place P5 in the next available slot from the left:
[P3, P4, P5, _, P6, P2, P7]
6. The only remaining part is P1. Place it in the remaining slot:
[P3, P4, P5, P1, P6, P2, P7]

Thus, the optimal processing sequence is:
P3 → P4 → P5 → P1 → P6 → P2 → P7

Step 3: Calculate the completion time for each job in sequence
Let us construct the schedule timeline:

1. Part P3:
• Milling: Starts at 0, processed for 3 hours, finishes at 3.
• Polishing: Starts at 3, processed for 4 hours, finishes at 3 + 4 = 7.

2. Part P4:
• Milling: Starts at 3, processed for 4 hours, finishes at 3 + 4 = 7.
• Polishing: Starts at max(7, 7) = 7, processed for 6 hours, finishes at 7 + 6 = 13.

3. Part P5:
• Milling: Starts at 7, processed for 5 hours, finishes at 7 + 5 = 12.
• Polishing: Starts at max(12, 13) = 13, processed for 7 hours, finishes at 13 + 7 = 20.

4. Part P1:
• Milling: Starts at 12, processed for 8 hours, finishes at 12 + 8 = 20.
• Polishing: Starts at max(20, 20) = 20, processed for 6 hours, finishes at 20 + 6 = 26.

5. Part P6:
• Milling: Starts at 20, processed for 6 hours, finishes at 20 + 6 = 26.
• Polishing: Starts at max(26, 26) = 26, processed for 4 hours, finishes at 26 + 4 = 30.

6. Part P2:
• Milling: Starts at 26, processed for 3 hours, finishes at 26 + 3 = 29.
• Polishing: Starts at max(29, 30) = 30, processed for 2 hours, finishes at 30 + 2 = 32.

7. Part P7:
• Milling: Starts at 29, processed for 2 hours, finishes at 29 + 2 = 31.
• Polishing: Starts at max(31, 32) = 32, processed for 1 hour, finishes at 32 + 1 = 33.

The final part P7 finishes all operations on the polishing machine at 33 hours. Therefore, the minimum total completion time is 33 hours.