Chapter 9, Problem 30AP

Summary Introduction

Interpretation: The performance according to sequencing rules FCFS, SPT, EDD and CR in relation to minimizing average tardiness has to be evaluated.

Concept Introduction:

Optimal sequence: It is supposed to be sequence or series of jobs that has to take place to minimize the idle time and at the same time all the jobs are completed.

Answer to Problem 30AP

When all the 4 sequencing rules are compared, it is found that FCFS rule gives minimum value of average tardiness while CR rule give the highest value of average tardiness. The values given by SPD and EDD rules are less than values provided by CR rule and more than value given by FCFS rule.

Explanation of Solution

Given information:

The information given to us is as follows:

Cars	Repair time (days)	Promised date
1	3	5
2	2	6
3	1	9
4	4	11
5	5	8

Let us know calculate the mean flow time and average tardiness.

  $\text{Tardiness = Completion time – Promising time}$

  $\text{Average tardiness= }\frac{\text{Total Tardiness}}{\text{Total number of jobs}}$

An important fact to be kept in mind is that in a situation where the due time is greater than the completion time, the tardiness should be considered as 0.

Calculation of average tardiness and number of tardy jobs using FCFS rule:

FCFS rule states that the person or job coming first will be attended first.

Cars	Repair time (days)	Completion time	Promised date	Tardiness
(1)	(2)	(3)	(4)	$\left(5\right)=\left(3\right)-\left(4\right)$
1	3	$\begin{array}{l}0+3=3\hfill \end{array}$	5	0
2	2	$3+2=5$	6	0
3	1	$5+1=6$	9	0
4	4	$\begin{array}{l}6+4=10\hfill \end{array}$	11	0
5	5	$10+5=15$	8	7
Total		$39$		7

Completion time is calculated by considering the cumulative of repair time values.

Since, we have derived the tardiness value; let us now calculate the average tardiness.

Calculation of Average Tardiness:

  $\text{Average tardiness= }\frac{\text{Total Tardiness}}{\text{Total number of jobs}}$

  $=\frac{7}{5}=1.4days$

Therefore, the average tardiness according to FCFS rule of sequencing is 1.4 days and the total number of tardy jobs is 1.

Calculation of average tardiness and number of tardy jobs using SPT rule:

SPT when expanded becomes shortest processing time. In the rule, the jobs must be arranged in increasing order according to the processing time.

Cars	Repair time (days)	Ranking done in increasing order
(1)	(2)	(3)
1	3	3
2	2	2
3	1	1
4	4	4
5	5	5
Total

After arranging the repair time in increasing order, the corresponding values in the Car column have to be considered for further calculations.

Cars	Repair time after ranking (days)	Completion time	Promised date	Tardiness
(1)	(2)	(3)	(4)	$\begin{array}{l}\left(5\right)=\left(3\right)-\left(4\right)\hfill \end{array}$
3	1	$0+1=1$	9	0
2	2	$\begin{array}{l}1+2=3\hfill \end{array}$	6	0
1	3	$\begin{array}{l}3+3=6\hfill \end{array}$	5	1
4	4	$\begin{array}{l}6+4=10\hfill \end{array}$	11	0
5	5	$\begin{array}{l}10+5=15\hfill \end{array}$	8	7
Total		35		8

Since, we have derived the tardiness value; let us now calculate the average tardiness.

Calculation of Average Tardiness:

  $\text{Average tardiness= }\frac{\text{Total Tardiness}}{\text{Total number of jobs}}$

  $=\frac{8}{5}=1.6days$

Therefore, the average tardiness according to SPT rule of sequencing is 1.6 days and the total number of tardy jobs is 2.

Calculation of average tardiness and number of tardy jobs using EDD rule:

EDD when expanded becomes Earliest due date. In the rule, the jobs must be arranged in increasing order according to the promised time.

Cars	Repair time (days)	Promised date	Ranking of promised date done in increasing order
1	3	5	1
2	2	6	2
3	1	9	4
4	4	11	5
5	5	8	3

After arranging the repair time in increasing order, the corresponding values in the Car column have to be considered for further calculations.

Cars (corresponding values according to ranks)	Repair time (days) (corresponding values according to ranks)	Completion time	Promised date after ranking	Tardiness
(1)	(2)	(3)	(4)	$\begin{array}{l}\left(5\right)=\left(3\right)-\left(4\right)\hfill \end{array}$
1	3	$0+3=3$	5	0
2	5	$3+2=5$	6	0
5	5	$\begin{array}{l}5+5=10\hfill \end{array}$	8	2
3	1	$\begin{array}{l}10+1=11\hfill \end{array}$	9	2
4	4	$11+4=15$	11	4
Total		44		8

Since, we have derived the tardiness value; let us now calculate the average tardiness.

Calculation of Average Tardiness:

  $\begin{array}{l}\text{Average tardiness= }\frac{\text{Total Tardiness}}{\text{Total number of jobs}}\\ \text{=}\frac{\text{8}}{\text{5}}\text{=1}\text{.6 days}\end{array}$

Therefore, the average tardiness according to EDD rule of sequencing is 1.6 days and the total number of tardy jobs is 3.

Finally, let us calculate the average tardiness and total number of tardy jobs according to CR rule.

CR when expanded becomes Critical Ratio. The formula to be used is as follows:

  $\text{Critical ratio=}\frac{\left(\text{Due time- Current time}\right)}{\text{Processing time}}$

Let us that the current time is 0.

Cars	Repair time (days)	Promised date	Critical ratio
1	3	5	$\frac{5-0}{3}=1.67$
2	2	6	$\frac{6-1}{2}=3$
3	1	9	$\frac{9-0}{1}=9$
4	4	11	$\frac{11-0}{4}=2.75$
5	5	8	$\frac{8-0}{5}=1.6$

Hence, the job having a minimum critical ration should be considered first. In our case, the car 5 has the minimum critical ratio of 1.6. So, it should be considered first for repair. After repair, car 5 can be ignored for further calculations.

Now, the current time for next critical ratio can be calculated as below:

  $\begin{array}{l}Currenttime=Currenttime+repairtimeofcurrentjob.\hfill \\ =0+5=5days.\hfill \end{array}$

Using this new current time, the critical ratio can be calculated for the rest of the cars.

Cars	Repair time (days)	Promised date	Critical ratio
1	3	5	$\frac{5-5}{3}=0$
2	2	6	$\frac{6-2}{2}=0.5$
3	1	9	$\frac{9-5}{1}=4$
4	4	11	$\frac{11-5}{4}=1.5$

Hence, the minimum critical ratio is 0 of Car 1. Hence car 1 should be considered for repair.

Further, a new current time has to be calculated.

  $\begin{array}{l}Currenttime=Currenttime+repairtimeofcurrentjob.\hfill \\ =5+3=8days.\hfill \end{array}$

By considering this value, we have to now calculate the critical ratio for the remaining 3 cars.

Cars	Repair time (days)	Promised date	Critical ratio
2	2	6	$\frac{6-8}{2}=-1$
3	1	9	$\frac{9-8}{1}=1$
4	4	11	$\frac{11-8}{4}=0.75$

Hence, car 2 has minimum critical ratio of -1. So, it has to be repaired after car1.

Further, a new current time has to be calculated for the remaining 2 cars

  $\begin{array}{l}Currenttime=Currenttime+repairtimeofcurrentjob.\hfill \\ =8+2=10days.\hfill \end{array}$

By considering this value, we have to now calculate the critical ratio for the remaining 3 cars.

Cars	Repair time (days)	Promised date	Critical ratio
3	1	9	$\frac{9-10}{1}=-1$
4	4	11	$\frac{11-10}{4}=0.25$

Hence, car 3 has minimum critical ratio of -1. So, it has to be repaired after car 2 and the remaining job i.e., repair of car 4 should be done last.

Therefore, the sequence we get using Critical ratio rule is $5-1-2-3-4.$

After deriving the sequence, we have to calculate the completion time and tardiness.

Calculation of completion time and tardiness:

Cars	Repair time (days)	Completion time	Promised date	Tardiness
(1)	(2)	(3)	(4)	$\begin{array}{l}\left(5\right)=\left(3\right)-\left(4\right)\hfill \end{array}$
5	5	$0+5=5$	8	0
1	3	$5+3=8$	5	3
2	2	$\begin{array}{l}8+2=10\hfill \end{array}$	6	4
3	1	$\begin{array}{l}10+1=11\hfill \end{array}$	9	2
4	4	$\begin{array}{l}11+4=15\hfill \end{array}$	11	4
Total		49		13

Since, we have derived the tardiness value; let us now calculate the average tardiness.

Calculation of Average Tardiness:

  $\begin{array}{l}\text{Average tardiness= }\frac{\text{Total Tardiness}}{\text{Total number of jobs}}\\ \text{=}\frac{\text{13}}{\text{5}}\text{=2}\text{.6 days}\end{array}$

Therefore, the average tardiness according to CR rule of sequencing is 2.6 days and the total number of tardy jobs is 4.

When all the 4 sequencing rules are compared, we find that FCFS rule gives minimum value of average tardiness while CR rule give the highest value of average tardiness. The values given by SPD and EDD rules are less than values provided by CR rule and more than value given by FCFS rule.