Chapter 15, Problem 19P

a)

Summary Introduction

To determine: Sequence of jobs based on decision rule First Come First Served (FCFS).

Introduction: 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 19P

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

Explanation of Solution

Given information:

The following information has been given:

Job	Date order received	Production days needed	Due date
A	110	20	180
B	120	30	200
C	122	10	175
D	125	16	230
E	130	18	210

All the jobs are arrived on day 130.

Determine the sequence of the job using FCFS:

Job sequence	Processing time	Flow time	Due date	Start day	End day	Lateness
A	20	20	180	130	149
B	30	50	200	150	179
C	10	60	175	180	189	14
D	16	76	230	190	205
E	18	94	210	206	223	13
Total	94	300				27

Working note:

Duration and due date for the jobs has been given. Flow time is the cumulative value of the duration. Jobs arrival rate is given as 130.

End day of Job A:

Jobs arrival rate is given as 130. 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 149.

$\begin{array}{c}\text{End day of Job A}=\text{Start day}+\left(\text{Duration }-1\right)\\ =130+\left(20-1\right)\\ =149\end{array}$

End day of Job B:

Start day of Job B is the next day of the end day of Job A. 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 179.

$\begin{array}{c}\text{End day of Job B}=\text{Start day}+\left(\text{Duration }-1\right)\\ =150+\left(30-1\right)\\ =179\end{array}$

End day of Job C:

Start day of Job C is the next day of the end day of Job B. 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 189.

$\begin{array}{c}\text{End day of Job C}=\text{Start day}+\left(\text{Duration }-1\right)\\ =180+\left(10-1\right)\\ =189\end{array}$

End day of Job D:

Start day of Job D is the next day of the end day of Job C. 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 205.

$\begin{array}{c}\text{End day of Job D}=\text{Start day}+\left(\text{Duration }-1\right)\\ =190+\left(16-1\right)\\ =205\end{array}$

End day of Job E:

Start day of Job E is the next day of the end day of Job D. 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 223.

$\begin{array}{c}\text{End day of Job E}=\text{Start day}+\left(\text{Duration }-1\right)\\ =206+\left(18-1\right)\\ =223\end{array}$

Lateness of Job A, Job B, and Job D:

End day of Job A, Job B, and Job D is less than its respective due date. Hence, there would be 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 14.

$\begin{array}{c}\text{Lateness of Job C}=\text{End day}-\text{Due date}\\ =189-175\\ =14\end{array}$

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 13.

$\begin{array}{c}\text{Lateness of Job E}=\text{End day}-\text{Due date}\\ =223-210\\ =13\end{array}$

b)

Summary Introduction

To determine: Sequence of jobs based on decision rule Earliest Due Date (EDD).

Introduction: 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 going on from the earliest due date to the latest due date.

Answer to Problem 19P

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

Explanation of Solution

Given information:

The following information has been given:

Job	Date order received	Production days needed	Due date
A	110	20	180
B	120	30	200
C	122	10	175
D	125	16	230
E	130	18	210

All the jobs are arrived on day 130.

Determine the sequence of the job using EDD:

Job sequence	Processing time	Flow time	Due date	Start day	End day	Lateness
C	10	10	175	130	139
A	20	30	180	140	159
B	30	60	200	160	189
E	18	78	210	190	207
D	16	94	230	208	223
Total	94	272

Working note:

Duration and due date for the jobs have been given. Flow time is the cumulative value of the duration. Jobs arrival rate is given as 130. The job should be arranged based on due date. Due date should be assembled in the ascending order.

End day of Job C:

Jobs arrival rate is given as 130. 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 139.

$\begin{array}{c}\text{End day of Job C}=\text{Start day}+\left(\text{Duration }-1\right)\\ =130+\left(10-1\right)\\ =139\end{array}$

End day of Job A:

Start day of Job A is the next day of the end day of Job C. 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 159.

$\begin{array}{c}\text{End day of Job A}=\text{Start day}+\left(\text{Duration }-1\right)\\ =140+\left(20-1\right)\\ =159\end{array}$

End day of Job B:

Start day of Job B is the next day of the end day of Job A. 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 189.

$\begin{array}{c}\text{End day of Job B}=\text{Start day}+\left(\text{Duration }-1\right)\\ =160+\left(30-1\right)\\ =189\end{array}$

End day of Job E:

Start day of Job E is the next day of the end day of Job B. 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 207.

$\begin{array}{c}\text{End day of Job E}=\text{Start day}+\left(\text{Duration }-1\right)\\ =190+\left(18-1\right)\\ =207\end{array}$

End day of Job D:

Start day of Job D is the next day of the end day of Job E. 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 223.

$\begin{array}{c}\text{End day of Job D}=\text{Start day}+\left(\text{Duration }-1\right)\\ =208+\left(16-1\right)\\ =223\end{array}$

Lateness of Job A, Job B, Job C, Job D, and Job E:

End day of all the jobs is less than its respective due date. Hence, there would be no lateness.

c)

Summary Introduction

To determine: Sequence of jobs based on decision rule Shortest Processing Time (SPT).

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

Answer to Problem 19P

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

Explanation of Solution

Given information:

The following information has been given:

Job	Date order received	Production days needed	Due date
A	110	20	180
B	120	30	200
C	122	10	175
D	125	16	230
E	130	18	210

All the jobs are arrived on day 130.

Determine the sequence of the job using SPT:

Job sequence	Processing time	Flow time	Due date	Start day	End day	Lateness
C	10	10	175	130	139
D	16	26	230	140	155
E	18	44	210	156	173
A	20	64	180	174	193	13
B	30	94	200	194	223	23
Total	94	238				36

Working note:

Duration and due date for the jobs has been given. Flow time is the cumulative value of the duration. Jobs arrival rate is given as 130. The job should be arranged based on the duration (processing time). The duration should be assembled in the ascending order.

End day of Job C:

Jobs arrival rate is given as 130. 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 139.

$\begin{array}{c}\text{End day of Job C}=\text{Start day}+\left(\text{Duration }-1\right)\\ =130+\left(10-1\right)\\ =139\end{array}$

End day of Job D:

Start day of Job D is the next day of the end day of Job C. 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 155.

$\begin{array}{c}\text{End day of Job D}=\text{Start day}+\left(\text{Duration }-1\right)\\ =140+\left(16-1\right)\\ =155\end{array}$

End day of Job E:

Start day of Job E is the next day of the end day of Job D. 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 173.

$\begin{array}{c}\text{End day of Job E}=\text{Start day}+\left(\text{Duration }-1\right)\\ =156+\left(18-1\right)\\ =173\end{array}$

End day of Job A:

Start day of Job A is the next day of the end day of Job E. 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 193.

$\begin{array}{c}\text{End day of Job A}=\text{Start day}+\left(\text{Duration }-1\right)\\ =174+\left(20-1\right)\\ =193\end{array}$

End day of Job B:

Start day of Job B is the next day of the end day of Job A. 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 223.

$\begin{array}{c}\text{End day of Job B}=\text{Start day}+\left(\text{Duration }-1\right)\\ =194+\left(30-1\right)\\ =223\end{array}$

Lateness of Job C, Job D, and Job E:

End day of Job C, Job D, and Job E are less than its respective due date. Hence, there would be no lateness.

Lateness of Job A:

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

$\begin{array}{c}\text{Lateness of Job A}=\text{End day}-\text{Due date}\\ =193-180\\ =13\end{array}$

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 23.

$\begin{array}{c}\text{Lateness of Job B}=\text{End day}-\text{Due date}\\ =223-200\\ =23\end{array}$

d)

Summary Introduction

To determine: Sequence of jobs based on decision rule Longest Processing Time (LPT).

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

Answer to Problem 19P

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

Explanation of Solution

Given information:

The following information has been given:

Job	Date order received	Production days needed	Due date
A	110	20	180
B	120	30	200
C	122	10	175
D	125	16	230
E	130	18	210

All the jobs are arrived on day 130.

Determine the sequence of the job using LPT:

Job sequence	Processing time	Flow time	Due date	Start day	End day	Lateness
B	30	30	200	130	159
A	20	50	180	160	179
E	18	68	210	180	197
D	16	84	230	198	213
C	10	94	175	214	223	48
Total	94	326				48

Working note:

Duration and due date for the jobs have been given. Flow time is the cumulative value of the duration. Jobs arrival rate is given as 130. The job should be arranged based on the duration (processing time). The duration should be assembled in the descending order

End day of Job B:

Jobs arrival rate is given as 130. 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 159.

$\begin{array}{c}\text{End day of Job B}=\text{Start day}+\left(\text{Duration }-1\right)\\ =130+\left(30-1\right)\\ =159\end{array}$

End day of Job A:

Start day of Job A is the next day of the end day of Job B. 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 179.

$\begin{array}{c}\text{End day of Job A}=\text{Start day}+\left(\text{Duration }-1\right)\\ =160+\left(20-1\right)\\ =179\end{array}$

End day of Job E:

Start day of Job E is the next day of the end day of Job A. 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 197.

$\begin{array}{c}\text{End day of Job E}=\text{Start day}+\left(\text{Duration }-1\right)\\ =180+\left(18-1\right)\\ =197\end{array}$

End day of Job D:

Start day of Job D is the next day of the end day of Job E. 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 213.

$\begin{array}{c}\text{End day of Job D}=\text{Start day}+\left(\text{Duration }-1\right)\\ =198+\left(16-1\right)\\ =213\end{array}$

End day of Job C:

Start day of Job C is the next day of the end day of Job D. 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 223.

$\begin{array}{c}\text{End day of Job C}=\text{Start day}+\left(\text{Duration }-1\right)\\ =214+\left(10-1\right)\\ =223\end{array}$

Lateness of Job B, Job A, Job E, and Job D:

End day of Job B, Job A, Job E, and Job D are less than its respective due date. Hence, there would be 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 48.

$\begin{array}{c}\text{Lateness of Job C}=\text{End day}-\text{Due date}\\ =223-175\\ =48\end{array}$

Summary:

Scheduling rule	Average lateness	Average flow time	Average number of jobs in system
FCFS	5.4	60.0	3.2
EDD	0.0	54.4	2.9
SPT	7.2	47.6	2.5
LPT	9.6	65.2	3.5

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 5.4.

$\begin{array}{c}\text{Average tartiness}=\frac{\text{Total lateness}}{\text{Number of jobs}}\\ =\frac{27}{5}\\ =5.4\end{array}$

Average flow time:

It is calculated by dividing the total flow time with the number of jobs. Hence, the average flow time is 60.

$\begin{array}{c}\text{Average flow time}=\frac{\text{Total flow time}}{\text{Number of jobs}}\\ =\frac{300}{5}\\ =60\end{array}$

Average number of jobs in the system:

It is calculated by dividing the total flow time with total duration of all the jobs. Hence, the average number of jobs in the system is 3.2.

$\begin{array}{c}\text{Average flow time}=\frac{\text{Total flow time}}{\text{Total duration}}\\ =\frac{300}{94}\\ =3.2\end{array}$

Earliest Due Date (EDD):

Average tardiness:

It is calculated by dividing the total lateness with the number of jobs. Hence, the average tardiness is 0.0.

$\begin{array}{c}\text{Average tartiness}=\frac{\text{Total lateness}}{\text{Number of jobs}}\\ =\frac{0}{5}\\ =0.0\end{array}$

Average flow time:

It is calculated by dividing the total flow time with the number of jobs. Hence, the average flow time is 54.4.

$\begin{array}{c}\text{Average flow time}=\frac{\text{Total flow time}}{\text{Number of jobs}}\\ =\frac{272}{5}\\ =54.4\end{array}$

Average number of jobs in the system:

It is calculated by dividing the total flow time with total duration of all the jobs. Hence, the average number of jobs in the system is 2.9.

$\begin{array}{c}\text{Average flow time}=\frac{\text{Total flow time}}{\text{Total duration}}\\ =\frac{272}{94}\\ =2.9\end{array}$

Shortest Processing Time (SPT):

Average tardiness:

It is calculated by dividing the total lateness with the number of jobs. Hence, the average tardiness is 7.2.

$\begin{array}{c}\text{Average tartiness}=\frac{\text{Total lateness}}{\text{Number of jobs}}\\ =\frac{36}{5}\\ =7.2\end{array}$

Average flow time:

It is calculated by dividing the total flow time with the number of jobs. Hence, the average flow time is 47.6.

$\begin{array}{c}\text{Average flow time}=\frac{\text{Total flow time}}{\text{Number of jobs}}\\ =\frac{238}{5}\\ =47.6\end{array}$

Average number of jobs in the system:

It is calculated by dividing the total flow time with total duration of all the jobs. Hence, the average number of jobs in the system is 2.5.

$\begin{array}{c}\text{Average flow time}=\frac{\text{Total flow time}}{\text{Total duration}}\\ =\frac{238}{94}\\ =2.5\end{array}$

Longest Processing Time (LPT):

Average tardiness:

It is calculated by dividing the total lateness with the number of jobs. Hence, the average tardiness is 9.6.

$\begin{array}{c}\text{Average tartiness}=\frac{\text{Total lateness}}{\text{Number of jobs}}\\ =\frac{48}{5}\\ =9.6\end{array}$

Average flow time:

It is calculated by dividing the total flow time with the number of jobs. Hence, the average flow time is 65.2.

$\begin{array}{c}\text{Average flow time}=\frac{\text{Total flow time}}{\text{Number of jobs}}\\ =\frac{326}{5}\\ =65.2\end{array}$

Average number of jobs in the system:

It is calculated by dividing the total flow time with total duration of all the jobs. Hence, the average number of jobs in the system is 3.55.

$\begin{array}{c}\text{Average flow time}=\frac{\text{Total flow time}}{\text{Total duration}}\\ =\frac{326}{94}\\ =3.5\end{array}$

Hence, Earliest Due Date is the best method for the average lateness. Shortest processing time has the minimum average flow time and average number of jobs in systems.