Chapter 6, Problem 6P

Twelve tasks, with times and precedence requirements as shown in the following table, are 10 be assigned to workstations using a cycle time of 1.5 minutes. Two heuristic rules will be tried: (1) greatest positional weight, and (2) most following tasks. In each case, the tiebreaker will be shortest processing time.

a. Draw the precedence diagram for this line.

b. Assign tasks to stations under each of the two rules.

c. Compute the percentage of idle tune fox each rule.

Summary Introduction

To draw: The precedence diagram.

Answer to Problem 6P

Precedence diagram:

Explanation of Solution

Given information:

Task	Duration (minutes)	Immediate (Predecessor)
a	0.1	Nil
b	0.2	a
c	0.9	b
d	0.6	c
e	0.1	Nil
f	0.2	d, e
g	0.4	f
h	0.1	g
i	0.2	h
j	0.7	i
k	0.3	j
l	0.2	k

Cycle time = 1.5 minutes

Precedence diagram:

The precedence diagram is drawn circles and arrows. The tasks are represented in circles and weights for each task are represented outside the circle. The arrows are represented to show which task is preceding the other task and so on.

Summary Introduction

To assign: Tasks on the basis of greatest positional weight.

Explanation of Solution

Given information:

Task	Duration (minutes)	Immediate (Predecessor)
a	0.1	Nil
b	0.2	a
c	0.9	b
d	0.6	c
e	0.1	Nil
f	0.2	d, e
g	0.4	f
h	0.1	g
i	0.2	h
j	0.7	i
k	0.3	j
l	0.2	k

Cycle time = 1.5 minutes

The number of following tasks, calculation of positional weight, processing time for each task is shown below.

Task	Following tasks	Number of following tasks	Calculation of positional weight	Positional weight	Processing time
a	b, c, d, f, g, h, i, j, k, l	10	0.1 + 0.2 + 0.9 + 0.6 + 0.2 + 0.4 + 0.1 + 0.2 + 0.7+ 0.3 + 0.2	3.9	0.1
b	c, d, f, g, h, i, j, k, l	9	0.2 + 0.9 + 0.6 + 0.2 + 0.4 + 0.1 + 0.2 + 0.7+ 0.3 + 0.2	3.8	0.2
c	d, f, g, h, i, j, k, l	8	0.9 + 0.6 + 0.2 + 0.4 + 0.1 + 0.2 + 0.7+ 0.3 + 0.2	3.6	0.9
d	f, g, h, i, j, k, l	7	0.6 + 0.2 + 0.4 + 0.1 + 0.2 + 0.7+ 0.3 + 0.2	2.7	0.6
e	f, g, h, i, j, k, l	7	0.1 + 0.2 + 0.4 + 0.1 + 0.2 + 0.7+ 0.3 + 0.2	2.2	0.1
f	g, h, i, j, k, l	6	0.2 + 0.4 + 0.1 + 0.2 + 0.7+ 0.3 + 0.2	2.1	0.2
g	h, i, j, k, l	5	0.4 + 0.1 + 0.2 + 0.7+ 0.3 + 0.2	1.9	0.4
h	i, j, k, l	4	0.1 + 0.2 + 0.7+ 0.3 + 0.2	1.5	0.1
i	j, k, l	3	0.2 + 0.7+ 0.3 + 0.2	1.4	0.2
j	k, l	2	0.7+ 0.3 + 0.2	1.2	0.7
k	l	1	0.3 + 0.2	0.5	0.3
l	Nil	0	0.2	0.2	0.2

Assigning tasks to workstations:

Workstation number	Eligible task	Assigned task	Task time	Unassigned cycle time	Reason
				1.5
1	a, e	a	0.1	1.4	Task 'a' has highest positional weight
	b, e	b	0.2	1.2	Task 'b' has highest positional weight
	c, e	c	0.9	0.3	Task 'c' has highest positional weight
	d, e	e	0.1	0.2	Task 'e' is the only eligible task available
	d	None		0.2 (Idle time)	The task time is greater than the unassigned cycle time.
				1.5
2	d	d	0.6	0.9	Task 'd' is the only eligible task available
	f	f	0.2	0.7	Task 'f' is the only eligible task available
	g	g	0.4	0.3	Task 'g' is the only eligible task available
	h	h	0.1	0.2	Task 'h' is the only eligible task available
	i	i	0.2	0	Task 'i' is the only eligible task available
				1.5
3	j	j	0.7	0.8	Task 'j' is the only eligible task available
	k	k	0.3	0.5	Task 'k' is the only eligible task available
	l	l	0.2	0.3	Task 'l' is the only eligible task available
				0.3 (Idle time)	All tasks completed

Overview of tasks assignment:

Workstation	Assigned tasks	Total cycle time used	Idle time
1	a, b, c, e	1.3	0.2
2	d, f, g, h, i	1.5	0
3	j, k, l	1.2	0.3

Summary Introduction

To assign: Tasks on the basis of most following tasks.

Explanation of Solution

Given information:

Task	Duration (minutes)	Immediate (Predecessor)
a	0.1	Nil
b	0.2	a
c	0.9	b
d	0.6	c
e	0.1	Nil
f	0.2	d, e
g	0.4	f
h	0.1	g
i	0.2	h
j	0.7	i
k	0.3	j
l	0.2	k

Cycle time = 1.5 minutes

The number of following tasks, calculation of positional weight, processing time for each task is shown below.

Task	Following tasks	Number of following tasks	Calculation of positional weight	Positional weight	Processing time
a	b, c, d, f, g, h, i, j, k, l	10	0.1 + 0.2 + 0.9 + 0.6 + 0.2 + 0.4 + 0.1 + 0.2 + 0.7+ 0.3 + 0.2	3.9	0.1
b	c, d, f, g, h, i, j, k, l	9	0.2 + 0.9 + 0.6 + 0.2 + 0.4 + 0.1 + 0.2 + 0.7+ 0.3 + 0.2	3.8	0.2
c	d, f, g, h, i, j, k, l	8	0.9 + 0.6 + 0.2 + 0.4 + 0.1 + 0.2 + 0.7+ 0.3 + 0.2	3.6	0.9
d	f, g, h, i, j, k, l	7	0.6 + 0.2 + 0.4 + 0.1 + 0.2 + 0.7+ 0.3 + 0.2	2.7	0.6
e	f, g, h, i, j, k, l	7	0.1 + 0.2 + 0.4 + 0.1 + 0.2 + 0.7+ 0.3 + 0.2	2.2	0.1
f	g, h, i, j, k, l	6	0.2 + 0.4 + 0.1 + 0.2 + 0.7+ 0.3 + 0.2	2.1	0.2
g	h, i, j, k, l	5	0.4 + 0.1 + 0.2 + 0.7+ 0.3 + 0.2	1.9	0.4
h	i, j, k, l	4	0.1 + 0.2 + 0.7+ 0.3 + 0.2	1.5	0.1
i	j, k, l	3	0.2 + 0.7+ 0.3 + 0.2	1.4	0.2
j	k, l	2	0.7+ 0.3 + 0.2	1.2	0.7
k	l	1	0.3 + 0.2	0.5	0.3
l	Nil	0	0.2	0.2	0.2

Assigning tasks to workstations:

Workstation number	Eligible task	Assigned task	Task time	Unassigned cycle time	Reason
				1.5
1	a, e	a	0.1	1.4	Task 'a' has the most following tasks
	b, e	b	0.2	1.2	Task 'b' has the most following tasks
	c, e	c	0.9	0.3	Task 'c' has the most following tasks
	d, e	e	0.1	0.2	Task 'e' is the only eligible task available
	d	None		0.2 (Idle time)	The task time is greater than the unassigned cycle time.
				1.5
2	d	d	0.6	0.9	Task 'd' is the only eligible task available
	f	f	0.2	0.7	Task 'f' is the only eligible task available
	g	g	0.4	0.3	Task 'g' is the only eligible task available
	h	h	0.1	0.2	Task 'h' is the only eligible task available
	i	i	0.2	0	Task 'i' is the only eligible task available
				1.5
3	j	j	0.7	0.8	Task 'j' is the only eligible task available
	k	k	0.3	0.5	Task 'k' is the only eligible task available
	l	l	0.2	0.3	Task 'l' is the only eligible task available
				0.3 (Idle time)	All tasks completed

Overview of tasks assignment:

Workstation	Assigned tasks	Total cycle time used	Idle time
1	a, b, c, e	1.3	0.2
2	d, f, g, h, i	1.5	0
3	j, k, l	1.2	0.3

Summary Introduction

To determine: The percentage of idle time.

Answer to Problem 6P

The percentage of idle time is 11.11%.

Explanation of Solution

Formula to calculate percentage of idle time:

$\text{Idle time}=\frac{\text{Total idle time}}{\text{Number of workstations}\times \text{Cycle time}}\times 100$

Calculation of percentage of idle time:

The solutions for most following tasks rule and highest positional weight is same. Hence, the percentage of idle time will also be the same.

$\begin{array}{c}\text{Idle time}=\frac{0.2+0.0+0.3}{3\times 1.5}\times 100\\ =\frac{0.5}{4.5}\times 100\\ =0.1111\times 100\\ =11.11\%\end{array}$

The percentage of idle time is 11.11%.