Chapter 16, Problem 7P

Using the information presented in the following table, identify the processing sequence that would result using (1) FCFS, (2) SPT, (3) EDD, and (4) CR. For each method, determine (1) average job flow time, (2) average job tardiness, and (3) average number of jobs in the system. Jobs are listed in order of arrival. (Hint: First determine the total job time for each job by computing the total processing time for the job and then adding in the setup time. All times and due dates are in hours.)

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 7P

The sequence is a-b-c-d-e.

Explanation of Solution

Given information:

Job	Processing time per unit	Units per job	Setup time	Due date
a	0.14	45	0.7	4
b	0.25	14	0.5	10
c	0.1	18	0.2	12
d	0.25	40	1	20
e	0.1	75	0.5	15

Due date is given. Job time should be determined using the given processing time per unit, units per job, and setup time.

Example:

$\begin{array}{c}\text{Job time}=\left(\text{Processing time}\times \text{Units per jobs}\right)+\text{Setup time}\\ =\left(0.14\times 45\right)+0.7\\ =7\text{ hours}\end{array}$

The procedure should be repeated for all the jobs. The resultant table is as follows:

Job	Job time (hours)	Due date (hours)
a	7	4
b	4	10
c	2	12
d	11	20
e	8	15

Determine the sequence using FCFS:

According to FCFS, the first come would be served first. Hence, the jobs should be sequenced in the order as per its arrival.

Hence, the sequence of jobs using FCFS is a-b-c-d-e.

Determine average flow time, average tardiness, and average number of jobs for FCFS:

Job	Processing time (hours)	Due date (hours)	Flow time	Tardiness
a	7	4	7	3
b	4	10	11	1
c	2	12	13	1
d	11	20	24	4
e	8	15	32	17
Total	32	61	87	26
Average	6.4	12.2	17.4	5.2

Supporting calculation:

Processing time and due date are given for each job. Flow time is the cumulative of the processing time.

Tardiness of Job a:

It is calculated by subtracting the flow time of the job from the due date of the project. Hence, the lateness of Job b is 3.

$\begin{array}{c}\text{Tardiness}=\text{Flow time}-\text{Due date}\\ =7-4\\ =3\end{array}$

Tardiness of Job b:

It is calculated by subtracting the flow time of the job from the due date of the project. Hence, the lateness of Job b is 1.

$\begin{array}{c}\text{Tardiness}=\text{Flow time}-\text{Due date}\\ =11-10\\ =1\end{array}$

Tardiness of Job c:

It is calculated by subtracting the flow time of the job from the due date of the project. Hence, the lateness of Job c is 1.

$\begin{array}{c}\text{Tardiness}=\text{Flow time}-\text{Due date}\\ =13-12\\ =1\end{array}$

Tardiness of Job d:

It is calculated by subtracting the flow time of the job from the due date of the project. Hence, the lateness of Job d is 4.

$\begin{array}{c}\text{Tardiness}=\text{Flow time}-\text{Due date}\\ =24-20\\ =4\end{array}$

Tardiness of Job e:

It is calculated by subtracting the flow time of the job from the due date of the project. Hence, the lateness of Job e is 17.

$\begin{array}{c}\text{Tardiness}=\text{Flow time}-\text{Due date}\\ =32-15\\ =17\end{array}$

Average flow time:

It is calculated by dividing the total flow time and number of jobs.

$\begin{array}{c}\text{Average flow time}=\frac{\text{Total flow time}}{\text{Number of jobs}}\\ =\frac{87}{5}\\ =17.40\text{ hours}\end{array}$

Hence, average flow time is 17.40 hours.

Average tardiness:

It is calculated by dividing the total tardiness and number of jobs.

$\begin{array}{c}\text{Average tardiness}=\frac{\text{Total tardiness}}{\text{Number of jobs}}\\ =\frac{26}{5}\\ =5.20\text{ hours}\end{array}$

Hence, average tardiness is 5.20 hours.

Average number of jobs:

It can be determined by dividing the total flow time and total processing time.

$\begin{array}{c}\text{Average number of jobs}=\frac{\text{Total flow time}}{\text{Total processing time}}\\ =\frac{87}{32}\\ =2.72\text{ jobs}\end{array}$

Hence, average number of jobs is 2.72 jobs.

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, job with the shortest duration would be served first. Then, the process would be going on from shortest to largest duration.

Answer to Problem 7P

The sequence is c-b-a-e-d.

Explanation of Solution

Given information:

Job	Processing time per unit	Units per job	Setup time	Due date
a	0.14	45	0.7	4
b	0.25	14	0.5	10
c	0.1	18	0.2	12
d	0.25	40	1	20
e	0.1	75	0.5	15

Due date is given. Job time should be determined using the given processing time per unit, units per job, and setup time.

Example:

$\begin{array}{c}\text{Job time}=\left(\text{Processing time}\times \text{Units per jobs}\right)+\text{Setup time}\\ =\left(0.14\times 45\right)+0.7\\ =7\text{ hours}\end{array}$

The procedure should be repeated for all the jobs. The resultant table is as follows:

Job	Job time (hours)	Due date (hours)
a	7	4
b	4	10
c	2	12
d	11	20
e	8	15

Determine the sequence using SPT:

According to SPT, the job that has the shortest processing would be served first and it goes on as the processing time increase. Duration should be assembled in the ascending order

Hence, the sequence of jobs using SPT is c-b-a-e-d.

Determine average flow time, average tardiness, and average number of jobs for SPT:

Job	Job time (hours)	Due date (hours)	Flow time	Tardiness
c	2	12	2	0
b	4	10	6	0
a	7	4	13	9
e	8	15	21	6
d	11	20	32	12
Total	32	61	74	27
Average	6.4	12.2	14.8	5.4

Supporting calculation:

Processing time and due date are given for each job. Flow time is the cumulative of the processing time.

Tardiness of Job c, and Job b:

Flow time of Job c and Job b is less than its respective due date. Hence, there would be no tardiness.

Tardiness of Job a:

It is calculated by subtracting the flow time of the job from the due date of the project. Hence, the lateness of Job a is 9.

$\begin{array}{c}\text{Tardiness}=\text{Flow time}-\text{Due date}\\ =13-4\\ =9\end{array}$

Tardiness of Job e:

It is calculated by subtracting the flow time of the job from the due date of the project. Hence, the lateness of Job e is 6.

$\begin{array}{c}\text{Tardiness}=\text{Flow time}-\text{Due date}\\ =21-15\\ =6\end{array}$

Tardiness of Job d:

It is calculated by subtracting the flow time of the job from the due date of the project. Hence, the lateness of Job d is 12.

$\begin{array}{c}\text{Tardiness}=\text{Flow time}-\text{Due date}\\ =32-20\\ =12\end{array}$

Average flow time:

It is calculated by dividing the total flow time and number of jobs.

$\begin{array}{c}\text{Average flow time}=\frac{\text{Total flow time}}{\text{Number of jobs}}\\ =\frac{74}{5}\\ =14.80\text{ hours}\end{array}$

Hence, average flow time is 14.80 hours.

Average tardiness:

It is calculated by dividing the total tardiness and number of jobs.

$\begin{array}{c}\text{Average tardiness}=\frac{\text{Total tardiness}}{\text{Number of jobs}}\\ =\frac{27}{5}\\ =5.40\text{ hours}\end{array}$

Hence, average tardiness is 5.40 hours.

Average number of jobs:

It can be determined by dividing the total flow time and total processing time.

$\begin{array}{c}\text{Average number of jobs}=\frac{\text{Total flow time}}{\text{Total processing time}}\\ =\frac{74}{32}\\ =2.31\text{ jobs}\end{array}$

Hence, average number of jobs is 2.31 jobs.

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, job with the earliest due date would be served first. Then, the process would be going on from earliest due date to latest due date.

Answer to Problem 7P

The sequence is a-b-c-e-d.

Explanation of Solution

Given information:

Job	Processing time per unit	Units per job	Setup time	Due date
a	0.14	45	0.7	4
b	0.25	14	0.5	10
c	0.1	18	0.2	12
d	0.25	40	1	20
e	0.1	75	0.5	15

Due date is given. Job time should be determined using the given processing time per unit, units per job, and setup time.

Example:

$\begin{array}{c}\text{Job time}=\left(\text{Processing time}\times \text{Units per jobs}\right)+\text{Setup time}\\ =\left(0.14\times 45\right)+0.7\\ =7\text{ hours}\end{array}$

The procedure should be repeated for all the jobs. The resultant table is as follows:

Job	Job time (hours)	Due date (hours)
a	7	4
b	4	10
c	2	12
d	11	20
e	8	15

Determine the sequence using EDD:

According to EDD, the job that has the earliest due date would be served first and it goes on as the due date increases. The job should be arranged based on due date. Due date should be assembled in the ascending order

Hence, the sequence of jobs using EDD is a-b-c-e-d.

Determine average flow time, average tardiness, and average number of jobs for EDD:

Job	Job time (hours)	Due date (hours)	Flow time	Tardiness
a	7	4	7	3
b	4	10	11	1
c	2	12	13	1
e	8	15	21	6
d	11	20	32	12
Total	32	61	84	23
Average	6.4	12.2	16.8	4.6

Supporting calculation:

Processing time and due date are given for each job. Flow time is the cumulative of the processing time.

Tardiness of Job a:

It is calculated by subtracting the flow time of the job from the due date of the project. Hence, the lateness of Job b is 3.

$\begin{array}{c}\text{Tardiness}=\text{Flow time}-\text{Due date}\\ =7-4\\ =3\end{array}$

Tardiness of Job b:

It is calculated by subtracting the flow time of the job from the due date of the project. Hence, the lateness of Job b is 1.

$\begin{array}{c}\text{Tardiness}=\text{Flow time}-\text{Due date}\\ =11-10\\ =1\end{array}$

Tardiness of Job c:

It is calculated by subtracting the flow time of the job from the due date of the project. Hence, the lateness of Job c is 1.

$\begin{array}{c}\text{Tardiness}=\text{Flow time}-\text{Due date}\\ =13-12\\ =1\end{array}$

Tardiness of Job e:

It is calculated by subtracting the flow time of the job from the due date of the project. Hence, the lateness of Job e is 6.

$\begin{array}{c}\text{Tardiness}=\text{Flow time}-\text{Due date}\\ =21-15\\ =6\end{array}$

Tardiness of Job d:

It is calculated by subtracting the flow time of the job from the due date of the project. Hence, the lateness of Job d is 12.

$\begin{array}{c}\text{Tardiness}=\text{Flow time}-\text{Due date}\\ =32-20\\ =12\end{array}$

Average flow time:

It is calculated by dividing the total flow time and number of jobs.

$\begin{array}{c}\text{Average flow time}=\frac{\text{Total flow time}}{\text{Number of jobs}}\\ =\frac{84}{5}\\ =16.80\text{ hours}\end{array}$

Hence, average flow time is 16.80 hours.

Average tardiness:

It is calculated by dividing the total tardiness and number of jobs.

$\begin{array}{c}\text{Average tardiness}=\frac{\text{Total tardiness}}{\text{Number of jobs}}\\ =\frac{23}{5}\\ =4.60\text{ hours}\end{array}$

Hence, average tardiness is 4.60 hours.

Average number of jobs:

It can be determined by dividing the total flow time and total processing time.

$\begin{array}{c}\text{Average number of jobs}=\frac{\text{Total flow time}}{\text{Total processing time}}\\ =\frac{84}{32}\\ =2.63\text{ jobs}\end{array}$

Hence, average number of jobs is 2.63 jobs.

Summary Introduction

To determine: Sequence of jobs based on decision rule critical ratio.

Introduction: Critical ratio is kind of scheduling rule that helps to identify that, the task or job is on the correct track. It would help to identify if the task is behind or ahead of the schedule.

Answer to Problem 7P

The sequence is a-b-c-e-d.

Explanation of Solution

Given information:

Job	Processing time per unit	Units per job	Setup time	Due date
a	0.14	45	0.7	4
b	0.25	14	0.5	10
c	0.1	18	0.2	12
d	0.25	40	1	20
e	0.1	75	0.5	15

Due date is given. Job time should be determined using the given processing time per unit, units per job, and setup time.

Example:

$\begin{array}{c}\text{Job time}=\left(\text{Processing time}\times \text{Units per jobs}\right)+\text{Setup time}\\ =\left(0.14\times 45\right)+0.7\\ =7\text{ hours}\end{array}$

The procedure should be repeated for all the jobs. The resultant table is as follows:

Job	Job time (hours)	Due date (hours)
a	7	4
b	4	10
c	2	12
d	11	20
e	8	15

Determine the sequence using critical ratio:

Initial critical ratio should be determined at day 0:

Job	Processing time (hours)	Due date	Critical ratio
a	7	4	0.57
b	4	10	2.5
c	2	12	6
d	11	20	1.82
e	8	15	1.88

Critical ratio for Job a:

It is can be determined by dividing the value attained by subtracting the completion day of previous job from the due date of current job with the processing time.

$\begin{array}{c}\text{Critical ratio}=\frac{\text{Due date}-\text{Completion day of completed job}}{\text{Processing time}}\\ =\frac{4-0}{7}\\ =0.57\end{array}$

Note: Process continues for all the jobs.

Job a has the lowest critical ratio. Thus, it will be completed first. Hence, Job a would be completed first in the sequence of jobs.

Determine the critical ratio after the completion of Job a:

As the processing time of job a is 7 hours, completion day of completed day would be 7.

Job	Processing time (hours)	Due date	Critical ratio
a	-	-	-
b	4	10	0.75
c	2	12	2.5
d	11	20	1.18
e	8	15	1

Critical ratio for Job b:

It is can be determined by dividing the value attained by subtracting the completion day of completed job from the due date of current job with the processing time.

$\begin{array}{c}\text{Critical ratio}=\frac{\text{Due date}-\text{Completion day of completed job}}{\text{Processing time}}\\ =\frac{10-7}{4}\\ =0.75\end{array}$

Note: Process continues for all the jobs.

Job b has the lowest critical ratio. Hence, Job b would be completed next in the sequence of jobs.

Determine the critical ratio after the completion of Job a and Job b:

As the processing time of job a is 7 hours and Job b is 4, completion day of completed day would be 11 (7+4).

Job	Processing time (hours)	Due date	Critical ratio
a	-	-	-
b	-	-	-
c	2	12	0.5
d	11	20	0.82
e	8	15	0.5

Critical ratio for Job c:

It is can be determined by dividing the value attained by subtracting the completion day of completed job from the due date of current job with the processing time.

$\begin{array}{c}\text{Critical ratio}=\frac{\text{Due date}-\text{Completion day of completed job}}{\text{Processing time}}\\ =\frac{12-11}{2}\\ =0.50\end{array}$

Job c has the lowest critical ratio (break the tie arbitrarily). Hence, Job c would be completed next in the sequence of jobs after Job a and Job b.

Determine the critical ratio after the completion of Job a, Job b and Job c:

As the processing time of job a is 7 hours, Job b is 4, and Job c is 2 hours. completion day of completed day would be 13 (7+4+2).

Job	Processing time (hours)	Due date	Critical ratio
a	-	-	-
b	-	-	-
c	-	-	-
d	11	20	0.82
e	8	15	0.5

Critical ratio for Job d:

It is can be determined by dividing the value attained by subtracting the completion day of completed job from the due date of current job with the processing time.

$\begin{array}{c}\text{Critical ratio}=\frac{\text{Due date}-\text{Completion day of completed job}}{\text{Processing time}}\\ =\frac{20-13}{11}\\ =0.64\end{array}$

Job e has the lowest critical ratio. Hence, Job e would be completed next in the sequence of jobs after Job a, Job b, and Job c.

As Job d is the remaining job, it will be completed next.

Hence, the sequence of jobs using critical ratio is a-b-c-e-d.

Average flow time:

It is calculated by dividing the total flow time and number of jobs.

$\begin{array}{c}\text{Average flow time}=\frac{\text{Total flow time}}{\text{Number of jobs}}\\ =\frac{84}{5}\\ =16.80\text{ hours}\end{array}$

Hence, average flow time is 16.80 hours.

Average tardiness:

It is calculated by dividing the total tardiness and number of jobs.

$\begin{array}{c}\text{Average tardiness}=\frac{\text{Total tardiness}}{\text{Number of jobs}}\\ =\frac{23}{5}\\ =4.60\text{ hours}\end{array}$

Hence, average tardiness is 4.60 hours.

Average number of jobs:

It can be determined by dividing the total flow time and total processing time.

$\begin{array}{c}\text{Average number of jobs}=\frac{\text{Total flow time}}{\text{Total processing time}}\\ =\frac{84}{32}\\ =2.63\text{ jobs}\end{array}$

Hence, average number of jobs is 2.63 jobs.