Chapter 17, Problem 12P

A project manager has compiled a list of major activities that will be required to install a computer information system in her firm. The list includes estimated completion times for activities and precedence relationships.

a. Construct a network diagram. You can use either AOA or AON (see Example 5).

b. If the project is finished within 26 weeks of its start, the project manager will receive a bonus of $1,000; and if the project is finished within 27 weeks of its start, the bonus will be $500. Find the probability of each bonus.

Summary Introduction

To construct: A network diagram.

Answer to Problem 12P

Answer

Network diagram:

Explanation of Solution

Given information:

Activity	Immediate predecessor	Optimistic time	Most likely time	Pessimistic time
A		2	4	6
D	A	6	8	10
E	D	7	9	12
H	E	2	3	5
F	A	3	4	8
G	F	5	7	9
B		2	2	3
I	B	2	3	6
J	I	3	4	5
K	J	4	5	8
C		5	8	12
M	C	1	1	1
N	M	6	7	11
O	N	8	9	13
End	H, G, K, O

Formula to calculate expected time and variance:

$\text{Expected time}=\frac{\text{Optimistic time}+\left(4\times \text{Most likely time}\right)+\text{Pessimistic time}}{6}$

$\text{Standard deviation}=\frac{\text{Pessimistic time}-\text{Optimistic time}}{6}$

$\text{Variance}=\frac{{\left(\text{Pessimistic time}-\text{Optimistic time}\right)}^2}{6^2}$

Calculation of mean and variance:

Activity	Optimistic time	Most likely time	Pessimistic time	Expected time	Standard deviation	Variance
	A	B	C	(A+(4*B)+C)/6	(C-A)/6	(C-A)^2/6^2
A	2	4	6	4	0.667	0.444
D	6	8	10	8	0.667	0.444
E	7	9	12	9.17	0.833	0.694
H	2	3	5	3.17	0.500	0.250
F	3	4	8	4.5	0.833	0.694
G	5	7	9	7	0.667	0.444
B	2	2	3	2.17	0.167	0.028
I	2	3	6	3.33	0.667	0.444
J	3	4	5	4	0.333	0.111
K	4	5	8	5.33	0.667	0.444
C	5	8	12	8.17	1.167	1.361
M	1	1	1	1	0.000	0.000
N	6	7	11	7.5	0.833	0.694
O	8	9	13	9.5	0.833	0.694
End

Network diagram:

The activity on arrow diagram is constructed. The activities are mentioned on the arrows. The arrows in the diagram are mentioned in the precedence order. The arrows indicates the successor for each node.

Summary Introduction

To determine: The probability of each bonus.

Answer to Problem 12P

The probability of achieving a bonus of $1,000 is 0.4099. The probability of achieving a bonus of $500 is 0.6742.

Explanation of Solution

Given information:

• If the project is finished in 26 weeks the bonus is $1,000.
• If the project is finished in 27 weeks the bonus is $500.

Calculation of expected duration, variance and standard deviation for each path:

Path A-D-E-H:

$\begin{array}{c}\text{Expected duration}=4+8+9.17+3.17\\ =24.34\end{array}$

$\begin{array}{c}\text{Variance}=0.444+0.444+0.694+0.250\\ =1.832\end{array}$

$\begin{array}{c}\text{Standard deviation}=\sqrt{\text{Variance}}\\ =\sqrt{1.832}\\ =1.3535\\ =1.354\end{array}$

Path A-F-G:

$\begin{array}{c}\text{Expected duration}=4+4.5+7\\ =15.50\end{array}$

$\begin{array}{c}\text{Variance}=0.444+0.694+0.444\\ =1.582\end{array}$

$\begin{array}{c}\text{Standard deviation}=\sqrt{\text{Variance}}\\ =\sqrt{1.582}\\ =1.2577\\ =1.258\end{array}$

Path B-I-J-K:

$\begin{array}{c}\text{Expected duration}=2.17+3.33+4+5.33\\ =14.83\end{array}$

$\begin{array}{c}\text{Variance}=0.028+0.444+0.111+0.444\\ =1.027\end{array}$

$\begin{array}{c}\text{Standard deviation}=\sqrt{\text{Variance}}\\ =\sqrt{1.027}\\ =1.014\end{array}$

Path C-M-N-O:

$\begin{array}{c}\text{Expected duration}=8.17+1+7.50+9.50\\ =26.17\end{array}$

$\begin{array}{c}\text{Variance}=1.361+0.0+0.694+0.694\\ =2.749\end{array}$

$\begin{array}{c}\text{Standard deviation}=\sqrt{\text{Variance}}\\ =\sqrt{2.749}\\ =1.658\end{array}$

Bonus of $1,000 (less than or equal to 26 weeks):

Calculation of probability of completion in less than or equal to 26 weeks:

Calculation of z value for all paths:

Formula:

$\text{Z}=\frac{\text{Specified time}-\text{Path mean}}{\text{Path standard deviation}}$

Path A-D-E-H:

$\begin{array}{c}\text{Z}=\frac{\text{26}-\text{24}\text{.34}}{\text{1}\text{.354}}\\ =1.23\end{array}$

From the standard normal distribution table,

The probability value for (z = 1.23) is 0.8907.

Path A-F-G:

$\begin{array}{c}\text{Z}=\frac{\text{26}-\text{15}\text{.50}}{\text{1}\text{.258}}\\ =8.35\end{array}$

Since z value is greater than +3.00, probability of completion is 1.0000.

Path B-I-J-K:

$\begin{array}{c}\text{Z}=\frac{\text{26}-\text{14}\text{.83}}{\text{1}\text{.014}}\\ =11.02\end{array}$

Since z value is greater than +3.00, probability of completion is 1.0000.

Path C-M-N-O:

$\begin{array}{c}\text{Z}=\frac{\text{26}-\text{26}\text{.17}}{\text{1}\text{.658}}\\ =-0.10\end{array}$

From the standard normal distribution table,

The probability value for (z = -0.10) is 0.4602.

Probability of completion in less than or equal to 26 weeks:

$\begin{array}{c}\text{Probability}=0.8907\times 1.0000\times 1.0000\times 0.4602\\ =0.4099\end{array}$

The probability of achieving a bonus of $1,000 is 0.4099.

Bonus of $500 (less than or equal to 27 weeks):

Calculation of probability of completion in less than or equal to 27 weeks:

Calculation of z value for all paths:

Formula:

$\text{Z}=\frac{\text{Specified time}-\text{Path mean}}{\text{Path standard deviation}}$

Path A-D-E-H:

$\begin{array}{c}\text{Z}=\frac{\text{27}-\text{24}\text{.34}}{\text{1}\text{.354}}\\ =1.96\end{array}$

From the standard normal distribution table,

The probability value for (z = 1.96) is 0.9750.

Path A-F-G:

$\begin{array}{c}\text{Z}=\frac{\text{27}-\text{15}\text{.50}}{\text{1}\text{.258}}\\ =9.14\end{array}$

Since z value is greater than +3.00, probability of completion is 1.0000.

Path B-I-J-K:

$\begin{array}{c}\text{Z}=\frac{\text{27}-\text{14}\text{.83}}{\text{1}\text{.014}}\\ =12\end{array}$

Since z value is greater than +3.00, probability of completion is 1.0000.

Path C-M-N-O:

$\begin{array}{c}\text{Z}=\frac{\text{27}-\text{26}\text{.17}}{\text{1}\text{.658}}\\ =0.50\end{array}$

From the standard normal distribution table,

The probability value for (z = 0.50) is 0.6915.

Probability of completion in less than or equal to 27 weeks:

$\begin{array}{c}\text{Probability}=0.9750\times 1.0000\times 1.0000\times 0.6915\\ =0.6742\end{array}$

The probability of achieving a bonus of $500 is 0.6742.