Chapter 3, Problem 23P

a)

Summary Introduction

To determine:The probability of completing the project in 17 months.

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 23P

The probability that the project completion time is 17 months is 0.0228.

Explanation of Solution

Given information:

$\begin{array}{c}\text{Project}\text{completion}=\text{21}\text{months}\\ \text{Project}\text{variance}=\text{4}\text{months}\end{array}$

Calculation of probability of completing the project in 17 months:

The standard deviationis $\text{σ}=2$, given the variance is 4. Therefore the z value is:

$\begin{array}{c}\text{z}=\frac{17-21}{2}\\ =-2\end{array}$

Z is calculated by dividing the difference of 17 and 21 with 2 which gives -2. From the normal distribution tables z value -2 corresponds to 0.0228

Hence, the probability that the project completion time is 17 months is 0.0228.

b)

Summary Introduction

To determine:The probability of completing the project in 20 months.

Answer to Problem 23P

The probability that the project completion time is 20 months is 0.3085.

Explanation of Solution

Given information:

$\begin{array}{c}\text{Project}\text{completion}=\text{21}\text{months}\\ \text{Project}\text{variance}=\text{4}\text{months}\end{array}$

Calculation of probability of completing the project in 20 months:

Calculate the standard deviation $\sigma =2$, and compute the z value first:

$\begin{array}{c}z=\frac{21-20}{2}\\ =-0.5\end{array}$

Z is calculated by dividing the difference of 21 and 20 with 2 which gives -0.5. From the normal distribution tables z value -0.5 corresponds to 0.3085.

Use the normal distribution tables to calculate the probability

The probability of project completion in 20 months is 0.3085.

Hence, the probability that the project completion time is 20 months is 0.3085.

c)

Summary Introduction

To determine:The probability of completing the project in 23 months.

Answer to Problem 23P

The probability that the project completion time is 23 months is 0.8413.

Explanation of Solution

Given information:

$\begin{array}{c}\text{Project}\text{completion}=\text{21}\text{months}\\ \text{Project}\text{variance}=\text{4}\text{months}\end{array}$

Calculation of probability of completing the project in 23 months:

Calculate the standard deviation $\sigma =2$, and compute the z value first

$\begin{array}{c}\text{z}=\frac{23-21}{2}\\ =1\end{array}$

Z is calculated by dividing the difference of 23 and 21 with 2 which gives 1. From the normal distribution tables z value 1 corresponds to 0.8413.

Use the normal distribution tables to calculate the probability

The probability of project completion in 23 months is 0.8413.

Hence, the probability that the project completion time is 23 months is 0.8413.

d)

Summary Introduction

To determine:The probability of completing the project in 25 months.

Answer to Problem 23P

The probability that the project completion time is 25 months is 0.9772.

Explanation of Solution

Given information:

$\begin{array}{c}\text{Project}\text{completion}=\text{21}\text{months}\\ \text{Project}\text{variance}=\text{4}\text{months}\end{array}$

Calculation of probability of completing the project in 25 months:

Calculate the standard deviation $\sigma =2$, and compute the z value,

$\begin{array}{c}\text{z}=\frac{25-21}{2}\\ =2\end{array}$

Z is calculated by dividing the difference of 25 and 21 with 2 which gives 2. From the normal distribution tables z value 2 corresponds to 0.9772.

Use the normal distribution tables to calculate the probability

The probability of project completion in 25 months is 0.9772.

Hence, the probability that the project completion time is 25 months is 0.9772.

e)

Summary Introduction

To compute: The due date that yields a 95% chance of completion.

Answer to Problem 23P

The due date that yields a 95% chance of completionis 24.2 months.

Explanation of Solution

Given information:

$\begin{array}{c}\text{Project}\text{completion}=\text{21}\text{months}\\ \text{Project}\text{variance}=\text{4}\text{months}\end{array}$

Calculation of due date that yields a 95% chance of completion:

The z value from the normal distribution tables as shown below,

From the above tables, observe that the z value is 1.65.

Compute the due date that yields a 95% chance of completion.

$\begin{array}{l}=21\text{months}+\left(\text{1}\text{.6}\times \text{2}\text{months}\right)\\ =\text{24}\text{.2}\text{months}\end{array}$.

The due date is calculated by adding 21 months with the product of 1.6 and 2 which yields 24.2 months.

Hence, the due date that yields a 95% chance of completion is 24.2 months.