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The Hickory Company manufactures wooden desks. Management schedules overtime every weekend to reduce the backlog on the most popular models. The automatic routing machine is used to cut certain types of edges on the desktops. The following orders need to be scheduled for the routing machine:

\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 due dates reflect the need for the order to be at its next operation. Develop separate schedules by using the FCFS and EDD rules. Compare the schedules on the basis of average flow time and average past due hours.

Using the FCFS (First Come, First Served) decision rule for sequencing the orders, the order is:

\begin{tabular}{l|ccccc}
\hline
Sequence & 1 & 2 & 3 & 4 & 5 \\
\hline
Order & 2 & 1 & 5 & 4 & 3 \\
\hline
\end{tabular}



Answer :

GinnyAnswer
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.

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