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Sagot :
Certainly! Let’s consider and solve the given scheduling problem step by step.
### Develop Schedules Using FCFS (First Come, First Served) Rule
The FCFS rule processes orders in the order in which they arrived.
Given the orders and their times:
\begin{tabular}{cccc}
\hline Order & \begin{tabular}{c}
Time Since Order \\
Arrived (hours ago)
\end{tabular} & \begin{tabular}{c}
Estimated Machine \\
Time (hours)
\end{tabular} & \begin{tabular}{c}
Due Date \\
(hours from now)
\end{tabular} \\
\hline 1 & 1 & 9 & 20 \\
2 & 0 & 7 & 21 \\
3 & 6 & 8 & 12 \\
4 & 5 & 3 & 8 \\
5 & 3 & 12 & 18 \\
\hline
\end{tabular}
The FCFS sequence is:
\begin{tabular}{l|ccccc}
\hline Sequence & 1 & 2 & 3 & 4 & 5 \\
\hline Order & 2 & 1 & 5 & 4 & 3 \\
\hline
\end{tabular}
Now, we need to calculate the following for this sequence:
- Total Flow Time
- Average Flow Time
- Total Past Due Hours
- Average Past Due Hours
We define:
- Flow Time for an order as the elapsed time from when the machine begins processing the order until it completes.
- Due Difference as the difference between Flow Time and the adjusted due date (i.e., due date minus time since order arrived).
#### FCFS Calculations:
1. Order 2:
- Machine Time: 7 hours
- Flow Time: 7
- Due Date: 21 - 0 = 21
- Past Due: max(0, 7 - 21) = 0 hours
2. Order 1:
- Machine Time: 9 hours
- Flow Time: 7 + 9 = 16
- Due Date: 20 - 1 = 19
- Past Due: max(0, 16 - 19) = 0 hours
3. Order 5:
- Machine Time: 12 hours
- Flow Time: 16 + 12 = 28
- Due Date: 18 - 3 = 15
- Past Due: max(0, 28 - 15) = 13 hours
4. Order 4:
- Machine Time: 3 hours
- Flow Time: 28 + 3 = 31
- Due Date: 8 - 5 = 3
- Past Due: max(0, 31 - 3) = 28 hours
5. Order 3:
- Machine Time: 8 hours
- Flow Time: 31 + 8 = 39
- Due Date: 12 - 6 = 6
- Past Due: max(0, 39 - 6) = 33 hours
Summarizing FCFS results:
- Total Flow Time = 7 + 16 + 28 + 31 + 39 = 121
- Average Flow Time = 121 / 5 = 24.2
- Total Past Due Hours = 0 + 0 + 13 + 28 + 33 = 74
- Average Past Due Hours = 74 / 5 = 14.8
### Develop the Schedule Using EDD (Earliest Due Date) Rule
The EDD rule processes orders based on their due dates, starting with the earliest.
Ordered by due dates:
\begin{tabular}{cccc}
\hline Order & \begin{tabular}{c}
Time Since Order \\
Arrived (hours ago)
\end{tabular} & \begin{tabular}{c}
Estimated Machine \\
Time (hours)
\end{tabular} & \begin{tabular}{c}
Due Date \\
(hours from now)
\end{tabular} \\
\hline 4 & 5 & 3 & 8 \\
3 & 6 & 8 & 12 \\
5 & 3 & 12 & 18 \\
1 & 1 & 9 & 20 \\
2 & 0 & 7 & 21 \\
\hline
\end{tabular}
The EDD sequence is:
\begin{tabular}{l|ccccc}
\hline Sequence & 1 & 2 & 3 & 4 & 5 \\
\hline Order & 4 & 3 & 5 & 1 & 2 \\
\hline
\end{tabular}
#### EDD Calculations:
1. Order 4:
- Machine Time: 3 hours
- Flow Time: 3
- Due Date: 8 - 5 = 3
- Past Due: max(0, 3 - 3) = 0 hours
2. Order 3:
- Machine Time: 8 hours
- Flow Time: 3 + 8 = 11
- Due Date: 12 - 6 = 6
- Past Due: max(0, 11 - 6) = 5 hours
3. Order 5:
- Machine Time: 12 hours
- Flow Time: 11 + 12 = 23
- Due Date: 18 - 3 = 15
- Past Due: max(0, 23 - 15) = 8 hours
4. Order 1:
- Machine Time: 9 hours
- Flow Time: 23 + 9 = 32
- Due Date: 20 - 1 = 19
- Past Due: max(0, 32 - 19) = 13 hours
5. Order 2:
- Machine Time: 7 hours
- Flow Time: 32 + 7 = 39
- Due Date: 21 - 0 = 21
- Past Due: max(0, 39 - 21) = 18 hours
Summarizing EDD results:
- Total Flow Time = 3 + 11 + 23 + 32 + 39 = 108
- Average Flow Time = 108 / 5 = 21.6
- Total Past Due Hours = 0 + 5 + 8 + 13 + 18 = 44
- Average Past Due Hours = 44 / 5 = 8.8
### Comparison of FCFS and EDD Schedules
- FCFS:
- Total Flow Time = 121
- Average Flow Time = 24.2
- Total Past Due Hours = 74
- Average Past Due Hours = 14.8
- EDD:
- Total Flow Time = 108
- Average Flow Time = 21.6
- Total Past Due Hours = 44
- Average Past Due Hours = 8.8
The EDD schedule shows better performance metrics compared to the FCFS schedule, providing lower average flow times and significantly fewer past due hours. This indicates that EDD is a more efficient scheduling rule for reducing late orders and managing workflow.
### Develop Schedules Using FCFS (First Come, First Served) Rule
The FCFS rule processes orders in the order in which they arrived.
Given the orders and their times:
\begin{tabular}{cccc}
\hline Order & \begin{tabular}{c}
Time Since Order \\
Arrived (hours ago)
\end{tabular} & \begin{tabular}{c}
Estimated Machine \\
Time (hours)
\end{tabular} & \begin{tabular}{c}
Due Date \\
(hours from now)
\end{tabular} \\
\hline 1 & 1 & 9 & 20 \\
2 & 0 & 7 & 21 \\
3 & 6 & 8 & 12 \\
4 & 5 & 3 & 8 \\
5 & 3 & 12 & 18 \\
\hline
\end{tabular}
The FCFS sequence is:
\begin{tabular}{l|ccccc}
\hline Sequence & 1 & 2 & 3 & 4 & 5 \\
\hline Order & 2 & 1 & 5 & 4 & 3 \\
\hline
\end{tabular}
Now, we need to calculate the following for this sequence:
- Total Flow Time
- Average Flow Time
- Total Past Due Hours
- Average Past Due Hours
We define:
- Flow Time for an order as the elapsed time from when the machine begins processing the order until it completes.
- Due Difference as the difference between Flow Time and the adjusted due date (i.e., due date minus time since order arrived).
#### FCFS Calculations:
1. Order 2:
- Machine Time: 7 hours
- Flow Time: 7
- Due Date: 21 - 0 = 21
- Past Due: max(0, 7 - 21) = 0 hours
2. Order 1:
- Machine Time: 9 hours
- Flow Time: 7 + 9 = 16
- Due Date: 20 - 1 = 19
- Past Due: max(0, 16 - 19) = 0 hours
3. Order 5:
- Machine Time: 12 hours
- Flow Time: 16 + 12 = 28
- Due Date: 18 - 3 = 15
- Past Due: max(0, 28 - 15) = 13 hours
4. Order 4:
- Machine Time: 3 hours
- Flow Time: 28 + 3 = 31
- Due Date: 8 - 5 = 3
- Past Due: max(0, 31 - 3) = 28 hours
5. Order 3:
- Machine Time: 8 hours
- Flow Time: 31 + 8 = 39
- Due Date: 12 - 6 = 6
- Past Due: max(0, 39 - 6) = 33 hours
Summarizing FCFS results:
- Total Flow Time = 7 + 16 + 28 + 31 + 39 = 121
- Average Flow Time = 121 / 5 = 24.2
- Total Past Due Hours = 0 + 0 + 13 + 28 + 33 = 74
- Average Past Due Hours = 74 / 5 = 14.8
### Develop the Schedule Using EDD (Earliest Due Date) Rule
The EDD rule processes orders based on their due dates, starting with the earliest.
Ordered by due dates:
\begin{tabular}{cccc}
\hline Order & \begin{tabular}{c}
Time Since Order \\
Arrived (hours ago)
\end{tabular} & \begin{tabular}{c}
Estimated Machine \\
Time (hours)
\end{tabular} & \begin{tabular}{c}
Due Date \\
(hours from now)
\end{tabular} \\
\hline 4 & 5 & 3 & 8 \\
3 & 6 & 8 & 12 \\
5 & 3 & 12 & 18 \\
1 & 1 & 9 & 20 \\
2 & 0 & 7 & 21 \\
\hline
\end{tabular}
The EDD sequence is:
\begin{tabular}{l|ccccc}
\hline Sequence & 1 & 2 & 3 & 4 & 5 \\
\hline Order & 4 & 3 & 5 & 1 & 2 \\
\hline
\end{tabular}
#### EDD Calculations:
1. Order 4:
- Machine Time: 3 hours
- Flow Time: 3
- Due Date: 8 - 5 = 3
- Past Due: max(0, 3 - 3) = 0 hours
2. Order 3:
- Machine Time: 8 hours
- Flow Time: 3 + 8 = 11
- Due Date: 12 - 6 = 6
- Past Due: max(0, 11 - 6) = 5 hours
3. Order 5:
- Machine Time: 12 hours
- Flow Time: 11 + 12 = 23
- Due Date: 18 - 3 = 15
- Past Due: max(0, 23 - 15) = 8 hours
4. Order 1:
- Machine Time: 9 hours
- Flow Time: 23 + 9 = 32
- Due Date: 20 - 1 = 19
- Past Due: max(0, 32 - 19) = 13 hours
5. Order 2:
- Machine Time: 7 hours
- Flow Time: 32 + 7 = 39
- Due Date: 21 - 0 = 21
- Past Due: max(0, 39 - 21) = 18 hours
Summarizing EDD results:
- Total Flow Time = 3 + 11 + 23 + 32 + 39 = 108
- Average Flow Time = 108 / 5 = 21.6
- Total Past Due Hours = 0 + 5 + 8 + 13 + 18 = 44
- Average Past Due Hours = 44 / 5 = 8.8
### Comparison of FCFS and EDD Schedules
- FCFS:
- Total Flow Time = 121
- Average Flow Time = 24.2
- Total Past Due Hours = 74
- Average Past Due Hours = 14.8
- EDD:
- Total Flow Time = 108
- Average Flow Time = 21.6
- Total Past Due Hours = 44
- Average Past Due Hours = 8.8
The EDD schedule shows better performance metrics compared to the FCFS schedule, providing lower average flow times and significantly fewer past due hours. This indicates that EDD is a more efficient scheduling rule for reducing late orders and managing workflow.
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