Chapter 15, Problem 15P

a)

Summary Introduction

To determine: Thesequence of jobs based on the decision rule, i.e, First Come First Served (FCFS).

Introduction: The First Come First Served is the scheduling rule, which helps to arrange the sequence in the order. Here, the first come would be served first.

Answer to Problem 15P

The sequence is A-B-C-D-E.

Explanation of Solution

Given information:

The following information has been given below:

Job	Due date	Duration (days)
A	313	8
B	312	16
C	325	40
D	314	5
E	314	3

All the jobs have arrived on day 275.

Determine the sequence of the job by using FCFS:

Job	Due date	Duration (days)	Flow time	Start	End	Lateness
A	313	8	8	275	282
B	312	16	24	283	298
C	325	40	64	299	338	13
D	314	5	69	339	343	29
E	314	3	72	344	346	32
Total		72	237			74

Working note:

The duration and due date for the jobs has been given. The flow time is the cumulative value of the duration. The jobs arrival rate is given as 275.

End day of Job A:

The jobs arrival rate is given as 275. The end day is calculated by adding the start day with the value attained by subtracting 1 from the duration. Hence, the end day of Job A is 282.

$\begin{array}{c}\text{End day of Job A}=\text{Start day}+\left(\text{Duration }-1\right)\\ =275+\left(8-1\right)\\ =282\end{array}$

End day of Job B:

The start day of Job B is the next day of the end day of Job A. The end day is calculated by adding the start day with the value attained by subtracting 1 from the duration. Hence, the end day of Job B is 298.

$\begin{array}{c}\text{End day of Job B}=\text{Start day}+\left(\text{Duration }-1\right)\\ =283+\left(16-1\right)\\ =298\end{array}$

End day of Job C:

The start day of Job C is the next day of the end day of Job B. The end day is calculated by adding the start day with the value attained by subtracting 1 from the duration. Hence, the end day of Job C is 338.

$\begin{array}{c}\text{End day of Job C}=\text{Start day}+\left(\text{Duration }-1\right)\\ =299+\left(40-1\right)\\ =338\end{array}$

End day of Job D:

The start day of Job D is the next day of the end day of Job C. The end day is calculated by adding the start day with the value attained by subtracting 1 from the duration. Hence, the end day of Job D is 343.

$\begin{array}{c}\text{End day of Job D}=\text{Start day}+\left(\text{Duration }-1\right)\\ =339+\left(5-1\right)\\ =343\end{array}$

End day of Job E:

The start day of Job E is the next day of the end day of Job D. The end day is calculated by adding the start day with the value attained by subtracting 1 from the duration. Hence, the end day of Job E is 346.

$\begin{array}{c}\text{End day of Job E}=\text{Start day}+\left(\text{Duration }-1\right)\\ =344+\left(3-1\right)\\ =346\end{array}$

Lateness of Job A and Job B:

The end day of Job A and Job B is less than its respective due date. Hence, there is no lateness.

Lateness of Job C:

It is calculated by subtracting the end day of the job from the due date of the project. Hence, the lateness of Job C is 13.

$\begin{array}{c}\text{Lateness of Job C}=\text{End day}-\text{Due date}\\ =338-325\\ =13\end{array}$

Lateness of Job D:

It is calculated by subtracting the end day of the job from the due date of the project. Hence, the lateness of Job D is 29.

$\begin{array}{c}\text{Lateness of Job D}=\text{End day}-\text{Due date}\\ =343-314\\ =29\end{array}$

Lateness of Job E:

It is calculated by subtracting the end day of the job from the due date of the project. Hence, the lateness of Job E is 32.

$\begin{array}{c}\text{Lateness of Job E}=\text{End day}-\text{Due date}\\ =346-314\\ =32\end{array}$

b)

Summary Introduction

To determine: The sequence of jobs based on the decision rule, i.e, Earliest Due Date (EDD).

Introduction: The Earliest Due Date is the scheduling rule, which helps to arrange the sequence in the order. Here, the job with the earliest due date would be served first. Then, the process would be go on from the earliest due date to the latest due date.

Answer to Problem 15P

The sequence is B-A-D-E-C.

Explanation of Solution

Given information:

The following information has been given below:

Job	Due date	Duration (days)
A	313	8
B	312	16
C	325	40
D	314	5
E	314	3

All the jobs have arrived on day 275.

Determine the sequence of the job by using EDD:

Job	Due date	Duration (days)	Flow time	Start	End	Lateness
B	312	16	16	275	290
A	313	8	24	291	298
D	314	5	29	299	303
E	314	3	32	304	306
C	325	40	72	307	346	21
Total		72	173			21

Working note:

The duration and due date for the jobs have been given. The flow time is the cumulative value of the duration. The jobs arrival rate is given as 275. The job should be arranged based on the due date. The due date should be assembled in the ascending order.

End day of Job A:

The jobs arrival rate is given as 275. The end day is calculated by adding the start day with the value attained by subtracting 1 from the duration. Hence, the end day of Job A is 290.

$\begin{array}{c}\text{End day of Job A}=\text{Start day}+\left(\text{Duration }-1\right)\\ =275+\left(16-1\right)\\ =290\end{array}$

End day of Job B:

The start day of Job B is the next day of the end day of Job A. The end day is calculated by adding the start day with the value attained by subtracting 1 from the duration. Hence, the end day of Job B is 298.

$\begin{array}{c}\text{End day of Job B}=\text{Start day}+\left(\text{Duration }-1\right)\\ =291+\left(8-1\right)\\ =298\end{array}$

End day of Job C:

The start day of Job C is the next day of the end day of Job B. The end day is calculated by adding the start day with the value attained by subtracting 1 from the duration. Hence, the end day of Job C is 303.

$\begin{array}{c}\text{End day of Job C}=\text{Start day}+\left(\text{Duration }-1\right)\\ =299+\left(5-1\right)\\ =303\end{array}$

End day of Job D:

The start day of Job D is the next day of the end day of Job C. The end day is calculated by adding the start day with the value attained by subtracting 1 from the duration. Hence, the end day of Job D is 306.

$\begin{array}{c}\text{End day of Job D}=\text{Start day}+\left(\text{Duration }-1\right)\\ =304+\left(3-1\right)\\ =306\end{array}$

End day of Job E:

The start day of Job E is the next day of the end day of Job D. The end day is calculated by adding the start day with the value attained by subtracting 1 from the duration. Hence, the end day of Job E is 346.

$\begin{array}{c}\text{End day of Job E}=\text{Start day}+\left(\text{Duration }-1\right)\\ =307+\left(40-1\right)\\ =346\end{array}$

Lateness of Job A, Job B, Job C, and Job D is given below:

The end day of Job A, Job B, Job C, and Job dare less than its respective due date. Hence, there is no lateness.

Lateness of Job E:

It is calculated by subtracting the end day of the job from the due date of the project. Hence, the lateness of Job E is 21.

$\begin{array}{c}\text{Lateness of Job E}=\text{End day}-\text{Due date}\\ =346-325\\ =21\end{array}$

c)

Summary Introduction

To determine: The sequence of jobs based on the decision rule, i.e, Shortest Processing Time (SPT).

Introduction: The Shortest Processing Time is the scheduling rule, which helps to arrange the sequence in the order. Here, the job with the shortest duration is served first. Then, the process would be go on from the shortest to the largest duration.

Answer to Problem 15P

The sequence is E-D-A-B-C.

Explanation of Solution

Given information:

The following information has been given below:

Job	Due date	Duration (days)
A	313	8
B	312	16
C	325	40
D	314	5
E	314	3

All the jobs are arrived on day 275.

Determine the sequence of the job by using SPT:

Job	Due date	Duration (days)	Flow time	Start	End	Lateness
E	314	3	3	275	277
D	314	5	8	278	282
A	313	8	16	283	290
B	312	16	32	291	306
C	325	40	72	307	346	21
Total		72	131			21

Working note:

The duration and due date for the jobs have been given. The flow time is the cumulative value of the duration. The jobs arrival rate is given as 275. The job should be arranged based on the duration (processing time). The duration should be assembled in the ascending order.

End day of Job A:

The jobs arrival rate is given as 275. The end day is calculated by adding the start day with the value attained by subtracting 1 from the duration. Hence, the end day of Job A is 277.

$\begin{array}{c}\text{End day of Job A}=\text{Start day}+\left(\text{Duration }-1\right)\\ =275+\left(3-1\right)\\ =277\end{array}$

End day of Job B:

The start day of Job B is the next day of the end day of Job A. The end day is calculated by adding the start day with the value attained by subtracting 1 from the duration. Hence, the end day of Job B is 282.

$\begin{array}{c}\text{End day of Job B}=\text{Start day}+\left(\text{Duration }-1\right)\\ =278+\left(5-1\right)\\ =282\end{array}$

End day of Job C:

The start day of Job C is the next day of the end day of Job B. The end day is calculated by adding the start day with the value attained by subtracting 1 from the duration. Hence, the end day of Job C is 290.

$\begin{array}{c}\text{End day of Job C}=\text{Start day}+\left(\text{Duration }-1\right)\\ =283+\left(8-1\right)\\ =290\end{array}$

End day of Job D:

The start day of Job D is the next day of the end day of Job C. The end day is calculated by adding the start day with the value attained by subtracting 1 from the duration. Hence, the end day of Job D is 306.

$\begin{array}{c}\text{End day of Job D}=\text{Start day}+\left(\text{Duration }-1\right)\\ =291+\left(16-1\right)\\ =306\end{array}$

End day of Job E:

The start day of Job E is the next day of the end day of Job D. The end day is calculated by adding the start day with the value attained by subtracting 1 from the duration. Hence, the end day of Job E is 346.

$\begin{array}{c}\text{End day of Job E}=\text{Start day}+\left(\text{Duration }-1\right)\\ =307+\left(40-1\right)\\ =346\end{array}$

Lateness of Job A, Job B, Job C, and Job D is as follows:

The end day of Job A, Job B, Job C, and Job D are less than its respective due date. Hence, there is no lateness.

Lateness of Job E:

It is calculated by subtracting the end day of the job from the due date of the project. Hence, the lateness of Job E is 21.

$\begin{array}{c}\text{Lateness of Job E}=\text{End day}-\text{Due date}\\ =346-325\\ =21\end{array}$

d)

Summary Introduction

To determine: The sequence of jobs based on the decision rule, i.e, Longest Processing Time (LPT).

Introduction: The Largest Processing Tine is the scheduling rule, which helps to arrange the sequence in the order. Here, the job with the largest duration is served first. Then, the process would go on from the largest to the shortest duration.

Answer to Problem 15P

The sequence is C-B-A-D-E.

Explanation of Solution

Given information:

The following information has been given below:

Job	Due date	Duration (days)
A	313	8
B	312	16
C	325	40
D	314	5
E	314	3

All the jobs have arrived on day 275.

Determine the sequence of the job by using LPT:

Job	Due date	Duration (days)	Flow time	Start	End	Lateness
C	325	40	40	275	314
B	312	16	56	315	330	18
A	313	8	64	331	338	25
D	314	5	69	339	343	29
E	314	3	72	344	346	32
Total		72	301			104

Working note:

The duration and due date for the jobs have been given. The flow time is the cumulative value of the duration. The jobs arrival rate is given as 275. The job is arranged based on the duration (processing time). The duration should be assembled in the descending order

End day of Job A:

The jobs arrival rate is given as 275. The end day is calculated by adding the start day with the value attained by subtracting 1 from the duration. Hence, the end day of Job A is 314.

$\begin{array}{c}\text{End day of Job A}=\text{Start day}+\left(\text{Duration }-1\right)\\ =275+\left(40-1\right)\\ =314\end{array}$

End day of Job B:

The start day of Job B is the next day of the end day of Job A. The end day is calculated by adding the start day with the value attained by subtracting 1 from the duration. Hence, the end day of Job B is 330.

$\begin{array}{c}\text{End day of Job B}=\text{Start day}+\left(\text{Duration }-1\right)\\ =315+\left(16-1\right)\\ =330\end{array}$

End day of Job C:

The start day of Job C is the next day of the end day of Job B. The end day is calculated by adding the start day with the value attained by subtracting 1 from the duration. Hence, the end day of Job C is 338.

$\begin{array}{c}\text{End day of Job C}=\text{Start day}+\left(\text{Duration }-1\right)\\ =331+\left(8-1\right)\\ =338\end{array}$

End day of Job D:

The start day of Job D is the next day of the end day of Job C. The end day is calculated by adding the start day with the value attained by subtracting 1 from the duration. Hence, the end day of Job D is 343.

$\begin{array}{c}\text{End day of Job D}=\text{Start day}+\left(\text{Duration }-1\right)\\ =339+\left(5-1\right)\\ =343\end{array}$

End day of Job E:

The start day of Job E is the next day of the end day of Job D. The end day is calculated by adding the start day with the value attained by subtracting 1 from the duration. Hence, the end day of Job E is 346.

$\begin{array}{c}\text{End day of Job E}=\text{Start day}+\left(\text{Duration }-1\right)\\ =344+\left(3-1\right)\\ =346\end{array}$

Lateness of Job A:

The end day of Job A is less than its respective due date. Hence, there is no lateness.

Lateness of Job B:

It is calculated by subtracting the end day of the job from the due date of the project. Hence, the lateness of Job B is 18.

$\begin{array}{c}\text{Lateness of Job B}=\text{End day}-\text{Due date}\\ =330-312\\ =18\end{array}$

Lateness of Job C:

It is calculated by subtracting the end day of the job from the due date of the project. Hence, the lateness of Job C is 25.

$\begin{array}{c}\text{Lateness of Job C}=\text{End day}-\text{Due date}\\ =338-313\\ =25\end{array}$

Lateness of Job D:

It is calculated by subtracting the end day of the job from the due date of the project. Hence, the lateness of Job D is 29.

$\begin{array}{c}\text{Lateness of Job D}=\text{End day}-\text{Due date}\\ =343-314\\ =29\end{array}$

Lateness of Job E:

It is calculated by subtracting the end day of the job from the due date of the project. Hence, the lateness of Job E is 32.

$\begin{array}{c}\text{Lateness of Job E}=\text{End day}-\text{Due date}\\ =346-314\\ =32\end{array}$

Summary:

Decision rule	Average tardiness	Average flow time	Average number of jobs in system
FCFS	14.8	47.4	3.3
EDD	4.2	34.6	2.4
SPT	4.2	26.2	1.8
LPT	20.8	60.2	4.2

First Come First Served (FCFS):

Average tardiness:

It is calculated by dividing the total lateness with the number of jobs. Hence, the average tardiness is 14.8.

$\begin{array}{c}\text{Average tartiness}=\frac{\text{Total lateness}}{\text{Number of jobs}}\\ =\frac{74}{5}\\ =14.8\end{array}$

Average flow time:

It is calculated by dividing the total flow time with the number of jobs. Hence, the average flow time is 47.4.

$\begin{array}{c}\text{Average flow time}=\frac{\text{Total flow time}}{\text{Number of jobs}}\\ =\frac{237}{5}\\ =47.4\end{array}$

Average number of jobs in the system:

It is calculated by dividing the total flow time with the total duration of all the jobs. Hence, the average number of jobs in the system is 3.3.

$\begin{array}{c}\text{Average flow time}=\frac{\text{Total flow time}}{\text{Total duration}}\\ =\frac{237}{72}\\ =3.3\end{array}$

Earliest Due Date (EDD):

Average tardiness:

It is calculated by dividing the total lateness with the number of jobs. Hence, the average tardiness is 4.2.

$\begin{array}{c}\text{Average tartiness}=\frac{\text{Total lateness}}{\text{Number of jobs}}\\ =\frac{21}{5}\\ =4.2\end{array}$

Average flow time:

It is calculated by dividing the total flow time with the number of jobs. Hence, the average flow time is 34.6.

$\begin{array}{c}\text{Average flow time}=\frac{\text{Total flow time}}{\text{Number of jobs}}\\ =\frac{173}{5}\\ =34.6\end{array}$

Average number of jobs in the system:

It is calculated by dividing the total flow time with the total duration of all the jobs. Hence, the average number of jobs in the system is 2.4.

$\begin{array}{c}\text{Average flow time}=\frac{\text{Total flow time}}{\text{Total duration}}\\ =\frac{173}{72}\\ =2.4\end{array}$

Shortest processing time (SPT):

Average tardiness:

It is calculated by dividing the total lateness with the number of jobs. Hence, the average tardiness is 4.2.

$\begin{array}{c}\text{Average tartiness}=\frac{\text{Total lateness}}{\text{Number of jobs}}\\ =\frac{21}{5}\\ =4.2\end{array}$

Average flow time:

It is calculated by dividing the total flow time with the number of jobs. Hence, the average flow time is 26.2.

$\begin{array}{c}\text{Average flow time}=\frac{\text{Total flow time}}{\text{Number of jobs}}\\ =\frac{131}{5}\\ =26.2\end{array}$

Average number of jobs in the system:

It is calculated by dividing the total flow time with the total duration of all the jobs. Hence, the average number of jobs in the system is 1.8.

$\begin{array}{c}\text{Average flow time}=\frac{\text{Total flow time}}{\text{Total duration}}\\ =\frac{131}{72}\\ =1.8\end{array}$

Largest processing time (LPT):

Average tardiness:

It is calculated by dividing the total lateness with the number of jobs. Hence, the average tardiness is 20.8.

$\begin{array}{c}\text{Average tartiness}=\frac{\text{Total lateness}}{\text{Number of jobs}}\\ =\frac{104}{5}\\ =20.8\end{array}$

Average flow time:

It is calculated by dividing the total flow time with the number of jobs. Hence, the average flow time is 60.2.

$\begin{array}{c}\text{Average flow time}=\frac{\text{Total flow time}}{\text{Number of jobs}}\\ =\frac{301}{5}\\ =60.2\end{array}$

Average number of jobs in the system:

It is calculated by dividing the total flow time with the total duration of all the jobs. Hence, the average number of jobs in the system is 4.2.

$\begin{array}{c}\text{Average flow time}=\frac{\text{Total flow time}}{\text{Total duration}}\\ =\frac{301}{72}\\ =4.2\end{array}$

Conclusion

Shortest processing time has the minimum average tardiness, average flow time, and the average number of jobs in systems. Hence, SPT is the best decision rule.