Chapter 16, Problem 6P

The following table contains information concerning four jobs that are awaiting processing at a work center.

Sequence the jobs using (1) FCFS, (2) SPT, (3) EDD, and (4) CR. Assume the list is by order of arrival.

b. For each of the methods in part a, determine (1) the average job flow time, (2) the average tardiness, and (3) the average number of jobs at the work center.

c. Is one method superior to the others? Explain.

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 6P

The sequence is A-B-C-D.

Explanation of Solution

Given information:

Job	Processing time (days)	Due date (days)
A	14	20
B	10	16
C	7	15
D	6	17

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.

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 6P

The sequence is A-B-C-D.

Explanation of Solution

Given information:

Job	Processing time (days)	Due date (days)
A	14	20
B	10	16
C	7	15
D	6	17

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.

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 6P

The sequence is D-C-B-A.

Explanation of Solution

Given information:

Job	Processing time (days)	Due date (days)
A	14	20
B	10	16
C	7	15
D	6	17

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 D-C-B-A.

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 6P

The sequence is C-B-D-A.

Explanation of Solution

Given information:

Job	Processing time (days)	Due date (days)
A	14	20
B	10	16
C	7	15
D	6	17

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 C-B-D-A.

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 6P

The sequence is A-C-D-B.

Explanation of Solution

Given information:

Job	Processing time (days)	Due date (days)
A	14	20
B	10	16
C	7	15
D	6	17

Determine the sequence using critical ratio:

Initial critical ratio should be determined at day 0:

Job	Processing time (days)	Due date (days)	Critical ratio
A	14	20	1.43
B	10	16	1.6
C	7	15	2.14
D	6	17	2.83

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.

$\begin{array}{c}\text{Critical ratio}=\frac{\text{Due date}-\text{Completion day of completed job}}{\text{Processing time}}\\ =\frac{20-0}{14}\\ =1.43\end{array}$

Critical ratio for Job B:

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.

$\begin{array}{c}\text{Critical ratio}=\frac{\text{Due date}-\text{Completion day of completed job}}{\text{Processing time}}\\ =\frac{16-0}{10}\\ =1.60\end{array}$

Critical ratio for Job C:

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.

$\begin{array}{c}\text{Critical ratio}=\frac{\text{Due date}-\text{Completion day of completed job}}{\text{Processing time}}\\ =\frac{15-0}{7}\\ =2.14\end{array}$

Critical ratio for Job D:

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.

$\begin{array}{c}\text{Critical ratio}=\frac{\text{Due date}-\text{Completion day of completed job}}{\text{Processing time}}\\ =\frac{17-0}{6}\\ =2.83\end{array}$

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 14 days, completion day of completed day would be 14.

Job	Processing time (days)	Due date (days)	Critical ratio
A	-	-	-
B	10	16	0.20
C	7	15	0.14
D	6	17	0.50

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.

$\begin{array}{c}\text{Critical ratio}=\frac{\text{Due date}-\text{Completion day of completed job}}{\text{Processing time}}\\ =\frac{16-14}{10}\\ =0.20\end{array}$

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.

$\begin{array}{c}\text{Critical ratio}=\frac{\text{Due date}-\text{Completion day of completed job}}{\text{Processing time}}\\ =\frac{15-14}{7}\\ =0.14\end{array}$

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.

$\begin{array}{c}\text{Critical ratio}=\frac{\text{Due date}-\text{Completion day of completed job}}{\text{Processing time}}\\ =\frac{17-14}{6}\\ =0.50\end{array}$

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

Determine the critical ratio after the completion of Job A and Job C:

As the processing time of job A is 14 days and Job C is 7, completion day of completed day would be 21 (14+7).

Job	Processing time (days)	Due date (days)	Critical ratio
A	-	-	-
B	10	16	-0.50
C
D	6	17	-0.67

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.

$\begin{array}{c}\text{Critical ratio}=\frac{\text{Due date}-\text{Completion day of completed job}}{\text{Processing time}}\\ =\frac{16-21}{10}\\ =-0.50\end{array}$

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.

$\begin{array}{c}\text{Critical ratio}=\frac{\text{Due date}-\text{Completion day of completed job}}{\text{Processing time}}\\ =\frac{17-21}{6}\\ =-0.67\end{array}$

Job D has the lowest critical ratio. Hence, Job D would be completed next in the sequence of jobs after Job A and Job C.

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

Hence, the sequence of jobs using critical ratio is A-C-D-B.

Summary Introduction

To determine: Average flow time, average tardiness, and average number of jobs for each method.

Introduction: Rules that are framed for providing the order in which the jobs need to be performed are termed as priority rules. These rules are more suitable for process-oriented facilities. The main purpose of priority rules are on-time completion of jobs, efficient utilization of services, and the maximizing the customer services.

Answer to Problem 6P

The average flow time, average tardiness, and average number of jobs has been determined for each method.

Explanation of Solution

Given information:

Job	Processing time (days)	Due date (days)
A	14	20
B	10	16
C	7	15
D	6	17

First Come First Served:

Job	Processing time (days)	Due date (days)	Flow time	Tardiness
A	14	20	14	0
B	10	16	24	8
C	7	15	31	16
D	6	17	37	20
Total	37	68	106	44
Average	9.3	17.0	26.5	11.0

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:

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

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 8.

$\begin{array}{c}\text{Tardiness}=\text{Flow time}-\text{Due date}\\ =24-16\\ =8\end{array}$

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 16.

$\begin{array}{c}\text{Tardiness}=\text{Flow time}-\text{Due date}\\ =31-15\\ =16\end{array}$

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 20.

$\begin{array}{c}\text{Tardiness}=\text{Flow time}-\text{Due date}\\ =37-17\\ =20\end{array}$

Average flow time:

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

$\begin{array}{c}\text{Average flow time}=\frac{\text{Total flow time}}{\text{Number of jobs}}\\ =\frac{106}{4}\\ =26.5\text{ days}\end{array}$

Hence, average flow time is 26.5 days.

Average tardiness:

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

$\begin{array}{c}\text{Average tardiness}=\frac{\text{Total tardiness}}{\text{Number of jobs}}\\ =\frac{44}{4}\\ =11.0\text{ days}\end{array}$

Hence, average tardiness is 11.0 days

Average number of jobs:

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

$\begin{array}{c}\text{Average number of jobs}=\frac{\text{Total flow time}}{\text{Total processing time}}\\ =\frac{106}{37}\\ =2.86\text{ jobs}\end{array}$

Hence, average number of jobs is 2.86 jobs.

Shortest processing time:

Job	Processing time (days)	Due date (days)	Flow time	Tardiness
D	6	17	6
C	7	15	13
B	10	16	23	7
A	14	20	37	17
Total	37	68	79	24
Average	9.3	17.0	19.8	12.0

Supporting calculation:

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

Tardiness of Job D and Job C:

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

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 7.

$\begin{array}{c}\text{Tardiness}=\text{Flow time}-\text{Due date}\\ =23-16\\ =7\end{array}$

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 17.

$\begin{array}{c}\text{Tardiness}=\text{Flow time}-\text{Due date}\\ =37-20\\ =17\end{array}$

Average flow time:

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

$\begin{array}{c}\text{Average flow time}=\frac{\text{Total flow time}}{\text{Number of jobs}}\\ =\frac{79}{4}\\ =19.75\text{ days}\end{array}$

Hence, average flow time is 19.75 days.

Average tardiness:

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

$\begin{array}{c}\text{Average tardiness}=\frac{\text{Total tardiness}}{\text{Number of jobs}}\\ =\frac{24}{4}\\ =6.0\text{ days}\end{array}$

Hence, average tardiness is 6.0 days

Average number of jobs:

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

$\begin{array}{c}\text{Average number of jobs}=\frac{\text{Total flow time}}{\text{Total processing time}}\\ =\frac{79}{37}\\ =2.14\text{ jobs}\end{array}$

Hence, average number of jobs is 2.14 jobs.

Earliest Due Date:

Job	Processing time (days)	Due date (days)	Flow time	Tardiness
C	7	15	7
B	10	16	17	1
D	6	17	23	6
A	14	20	37	17
Total	37	68	84	24
Average	9.3	17.0	21.0	8.0

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:

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

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.

$\begin{array}{c}\text{Tardiness}=\text{Flow time}-\text{Due date}\\ =16-17\\ =1\end{array}$

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 6.

$\begin{array}{c}\text{Tardiness}=\text{Flow time}-\text{Due date}\\ =23-17\\ =6\end{array}$

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 17.

$\begin{array}{c}\text{Tardiness}=\text{Flow time}-\text{Due date}\\ =37-20\\ =17\end{array}$

Average flow time:

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

$\begin{array}{c}\text{Average flow time}=\frac{\text{Total flow time}}{\text{Number of jobs}}\\ =\frac{84}{4}\\ =21.00\text{ days}\end{array}$

Hence, average flow time is 21.00 days.

Average tardiness:

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

$\begin{array}{c}\text{Average tardiness}=\frac{\text{Total tardiness}}{\text{Number of jobs}}\\ =\frac{24}{4}\\ =6.0\text{ days}\end{array}$

Hence, average tardiness is 6.0 days

Average number of jobs:

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

$\begin{array}{c}\text{Average number of jobs}=\frac{\text{Total flow time}}{\text{Total processing time}}\\ =\frac{84}{37}\\ =2.27\text{ jobs}\end{array}$

Hence, average number of jobs is 2.27 jobs.

Critical ratio:

Job	Processing time (days)	Due date (days)	Flow time	Tardiness
A	14	20	14	0
C	7	15	21	6
D	6	17	27	10
B	10	16	37	21
Total	37	68	99	37
Average	9.3	17.0	24.8	9.3

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:

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

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 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 10.

$\begin{array}{c}\text{Tardiness}=\text{Flow time}-\text{Due date}\\ =27-17\\ =10\end{array}$

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 21.

$\begin{array}{c}\text{Tardiness}=\text{Flow time}-\text{Due date}\\ =37-16\\ =21\end{array}$

Average flow time:

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

$\begin{array}{c}\text{Average flow time}=\frac{\text{Total flow time}}{\text{Number of jobs}}\\ =\frac{99}{4}\\ =24.75\text{ days}\end{array}$

Hence, average flow time is 24.75 days.

Average tardiness:

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

$\begin{array}{c}\text{Average tardiness}=\frac{\text{Total tardiness}}{\text{Number of jobs}}\\ =\frac{37}{4}\\ =9.25\text{ days}\end{array}$

Hence, average tardiness is 9.25  days

Average number of jobs:

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

$\begin{array}{c}\text{Average number of jobs}=\frac{\text{Total flow time}}{\text{Total processing time}}\\ =\frac{99}{37}\\ =2.68\text{ jobs}\end{array}$

Hence, average number of jobs is 2.68 jobs.

Summary Introduction

To determine: The method which is superior to others.

Introduction: Rules that are framed for providing the order in which the jobs need to be performed are termed as priority rules. These rules are more suitable for process-oriented facilities. The main purpose of priority rules are on-time completion of jobs, efficient utilization of services, and the maximizing the customer services.

Answer to Problem 6P

SPT is superior to others.

Explanation of Solution

Given information:

Job	Processing time (days)	Due date (days)
A	14	20
B	10	16
C	7	15
D	6	17

Determine the method which is superior to other:

Shortest Processing Time (SPT) is the method which is superior to other, as it is the method which would provide lowest value when compare to others.