Chapter 3, Problem 19P

Kelle Carpet and Trim installs carpet in commercial offices. Peter Kelle has been very concerned with the amount of time it took to complete several recent jobs. Some of his workers are very unreliable. A list of activities and their optimistic completion time, the most likely completion time, and the pessimistic completion time (all in days) for a new contract are given in the following table:

a) Determine the expected completion time and variance for each activity.

b) Determine the total project completion time and the critical path for the project.

c) Determine ES, EF, LS, LF, and slack for each activity.

d) What is the probability that Kelle Carpet and Trim will finish the project in 40 days or less?

Summary Introduction

To determine:The expected completion time and variance of the various activities.

Introduction:

The activity times of various tasks in a PERT project network are assumed to follow a probability distribution. For calculating the expected times and standard deviation, the parameters required are the following:

• Optimistic time: This is the time denoted by “a”, which is the best possible or in other words, the quickest time in which an activity can be completed, assuming that everything goes as per plan.
• Pessimistic time: This is the worst case scenario, where all the conditions are adverse or unfavorable. The maximum time which may be taken in such a situation is the pessimistic time denoted by “b”.
• Most likely time: The most realistic estimate of the time, denoted by “m” in normal conditions, is another parameter which is important in the computation of expected times and variances.

Answer to Problem 19P

The expected times and standard deviations for the following six activities are shown in Table 1.

Explanation of Solution

Given information:

Activity	a	m	b	t	Immediate predecessor
A	3	6	8	8
B	2	4	4	4
C	1	2	3	3
D	6	7	8	8	C
E	2	4	6	6	B,D
F	6	10	14	14	A,E
G	1	2	4	4	A,E
H	3	6	9	9	F
I	10	11	12	12	G
J	14	16	20	20	C
K	2	8	10	10	H,I

Formula:

The expected time “t” would be calculated by the formula shown below.

$\text{t}=\frac{\left(\text{a}+\text{4m}+\text{b}\right)}{\text{6}}$

where “a” is the optimistic time, “b” is the pessimistic time and “m” is the most realistic time.

The variance denoted by σ² is calculated by using the formula shown below.

$\text{Variance}={\left(\frac{\text{b}-\text{a}}{\text{6}}\right)}^{\text{2}}$

Calculation of expected completion time and variance of the various activities:

Use the above formulas to obtain the expected completion time and the variance for the various activities, as shown below.

Unit=number of days

Activity	a	m	b	t	Immediate predecessor	t	SD	Variance
A	3	6	8	8		5.83	0.83	0.69
B	2	4	4	4		3.67	0.33	0.11
C	1	2	3	3		2.00	0.33	0.11
D	6	7	8	8	C	7.00	0.33	0.11
E	2	4	6	6	B,D	4.00	0.67	0.44
F	6	10	14	14	A,E	10.00	1.33	1.78
G	1	2	4	4	A,E	2.17	0.50	0.25
H	3	6	9	9	F	6.00	1.00	1.00
I	10	11	12	12	G	11.00	0.33	0.11
J	14	16	20	20	C	16.33	1.00	1.00
K	2	8	10	10	H,I	7.33		0.00

Table 1

Excel worksheet:

Hence, the expected times and standard deviations for the following six activities are shown in Table 1

Summary Introduction

To determine: The critical path and the project completion time.

Answer to Problem 19P

The longest path is CàDàEàFàHàK which is the critical path and the project completion time is 36.33 days.

Explanation of Solution

Given information:

Activity	a	m	b	t	Immediate predecessor
A	3	6	8	8
B	2	4	4	4
C	1	2	3	3
D	6	7	8	8	C
E	2	4	6	6	B,D
F	6	10	14	14	A,E
G	1	2	4	4	A,E
H	3	6	9	9	F
I	10	11	12	12	G
J	14	16	20	20	C
K	2	8	10	10	H,I

Calculation of critical path and the project completion time:

The AON project network diagram with the precedence relationships of various activities is shown below,

The various project paths from the above diagram and the path durations are in Table 1.

• AàFàHàK, path accounting for a duration of $5.83+10+6+7.33=29.16\text{days}$

• Path AàGàIàK accounting for a duration of $5.83+2.17+11+7.33=26.33\text{days}$

• Project path BàEàFàHàK accounting for a duration of $3.67+4+10+6+7.33=31\text{days}$

• Project path BàEàGàIàK accounting for a duration of $3.67+4+2.17+11+7.33=28.17\text{days}$.

• Project path CàDàEàGàIàK accounting for a duration of $2+7+4+2.17+11+7.33=33.50\text{days}$

• Project path CàDàEàFàHàK accounting for a duration of $2+7+4+2.17+10+6+7.33=36.33\text{days}$.

• Project path CàJ accounting for a duration of $2+16.33=18.33\text{days}$

The longest path is CàDàEàFàHàK and is the critical path. Therefore, the project completion time is 36.33days.

Hence, The critical path is CàDàEàFàHàK and the project completion time is 36.33 days.

Summary Introduction

To determine: Early start, Early finish, Late start, Late finish and slack for each activity.

Answer to Problem 19P

The Early start, Early finish, Late start, Late finish and slack for each activity is shown in Table 2.

Explanation of Solution

Given information:

Activity	a	m	b	t	Immediate predecessor
A	3	6	8	8
B	2	4	4	4
C	1	2	3	3
D	6	7	8	8	C
E	2	4	6	6	B,D
F	6	10	14	14	A,E
G	1	2	4	4	A,E
H	3	6	9	9	F
I	10	11	12	12	G
J	14	16	20	20	C
K	2	8	10	10	H,I

Calculation of Early start, Early finish, Late start, Late finish and slack for each activity:

The AON diagram along with the time duration for each project activity is shown below.

Slack activity can be calculated by finding the difference LS-ES or LF-EF. ES, EF, LS and LF for the project as shown in the table below.

Table 2

The slack for each activity is shown in the above table

CàDàEàFàHàK is the critical path and therefore activities C, D, E, F, H and K have no slack.

The slack in activity A is 7.17 days, the slack in activity B is 5.33 days, the slack in activity G is 2.83 days, the slack in activity I is 2.83 days and the slack in activity J is 18 days.

Hence, the Early start, Early finish, Late start, Late finish and slack for each activity is shown in Table 2.

Summary Introduction

To calculate:The probability that the project will be completed in 40 days or less.

Answer to Problem 19P

The probability that the project will be completed in 40 days or less is 94.63%.

Explanation of Solution

Given information:

Activity	a	m	b	t	Immediate predecessor
A	3	6	8	8
B	2	4	4	4
C	1	2	3	3
D	6	7	8	8	C
E	2	4	6	6	B,D
F	6	10	14	14	A,E
G	1	2	4	4	A,E
H	3	6	9	9	F
I	10	11	12	12	G
J	14	16	20	20	C
K	2	8	10	10	H,I

Calculation of probability that the project will be completed in 40 days or less:

Consider the expected project completion time of 36.33 days as the mean and the variance of the distribution σ², is the sum of the variances of individual activities on the critical path which are C, D, E, F, H, and K (refer table 1)

Consequently,

$\begin{array}{c}{\text{σ}}^{\text{2}}=0.11+0.11+0.44+1.78+1.00+1.78\\ =5.22\end{array}$

Therefore the standard deviation is the square root of 5.22, which works out to 2.28

Calculate the z value as shown below

$\begin{array}{c}\text{z}=\frac{40-36.33}{2.28}\\ =1.61\end{array}$

From thez-factor table1.61 corresponds to 0.9463. The probability that the project will be completed in 40 days is 94.63%.

Hence, the probability that the project will be completed in 40 days or less is 94.63%.