Chapter 18, Problem 14PA

(a)

Summary Introduction

Interpretation: The probability that the project is completed within $33\text{weeks}$ is to be determined.

Concept Introduction:

Project Scheduling is used in Project planning and coordination by using Program Evaluation Review Technique (PERT) and Critical Path Method (CPM). PERT is a network based scheduling procedure. This technique is used to plan, schedule and control different project such as construction works, design installation of system, maintenance work and research and development process.

Explanation of Solution

The activity with optimistic, most probable and pessimistic time is tabulated below.

Activity	Optimistic time	Most Probable Time	Pessimistic Time
A	$4$	$5$	$6$
B	$2.5$	$3$	$3.5$
C	$6$	$7$	$8$
D	$5$	$5.5$	$9$
E	$6$	$7$	$8$
F	$1$	$3$	$5$
G	$8$	$10$	$12$

Table (1)

The expression for the estimated time is given by,

  $t_e=\frac{t_0+4t_m+t_p}{6}$

Here, $t_e$ is the estimated time of an activity, $t_0$ is the optimistic time, $t_m$ is the most probable time and $t_p$ is the pessimistic time.

The expression for variance is given by,

  $\text{Variance}=\frac{{\left(t_p-t_0\right)}^2}{36}$

The variance and estimated time of each activity is tabulated below.

Activity	Optimistic time	Most Probable Time	Pessimistic Time	Estimated Time	Variance
A	$4$	$5$	$6$	$5$	$0.111$
B	$2.5$	$3$	$3.5$	$3$	$0.028$
C	$6$	$7$	$8$	$7$	$0.111$
D	$5$	$5.5$	$9$	$6$	$0.444$
E	$6$	$7$	$8$	$7$	$0.111$
F	$1$	$3$	$5$	$3$	$0.444$
G	$8$	$10$	$12$	$10$	$0.444$

Table (2)

Calculate $z$ value to find the probability of meeting a $33\text{weeks}$ deadline.

  $\begin{array}{c}z=\frac{33-32}{\sqrt{1.69}}\\ =\frac{1}{1.3}\\ =0.769\end{array}$

Refer probability distribution table.

For $z\le 0.769$ .

  $P\left(z\le 0.769\right)=0.77935$ .

Hence, the probability that the project is completed within $33\text{weeks}$ is $0.77935$ .

(b)

Summary Introduction

Interpretation: The probability that the project is completed within $30\text{weeks}$ is to be determined.

Concept Introduction:

Project Scheduling is used in Project planning and coordination by using Program Evaluation Review Technique (PERT) and Critical Path Method (CPM). PERT is a network based scheduling procedure. This technique is used to plan, schedule and control different project such as construction works, design installation of system, maintenance work and research and development process.

Explanation of Solution

The activity with optimistic, most probable and pessimistic time is tabulated below.

Activity	Optimistic time	Most Probable Time	Pessimistic Time
A	$4$	$5$	$6$
B	$2.5$	$3$	$3.5$
C	$6$	$7$	$8$
D	$5$	$5.5$	$9$
E	$6$	$7$	$8$
F	$1$	$3$	$5$
G	$8$	$10$	$12$

Table (1)

The expression for the estimated time is given by,

  $t_e=\frac{t_0+4t_m+t_p}{6}$

Here, $t_e$ is the estimated time of an activity, $t_0$ is the optimistic time, $t_m$ is the most probable time and $t_p$ is the pessimistic time.

The expression for variance is given by,

  $\text{Variance}=\frac{{\left(t_p-t_0\right)}^2}{36}$

The variance and estimated time of each activity is tabulated below.

Activity	Optimistic time	Most Probable Time	Pessimistic Time	Estimated Time	Variance
A	$4$	$5$	$6$	$5$	$0.111$
B	$2.5$	$3$	$3.5$	$3$	$0.028$
C	$6$	$7$	$8$	$7$	$0.111$
D	$5$	$5.5$	$9$	$6$	$0.444$
E	$6$	$7$	$8$	$7$	$0.111$
F	$1$	$3$	$5$	$3$	$0.444$
G	$8$	$10$	$12$	$10$	$0.444$

Table (2)

Calculate $z$ value to find the probability of meeting a $30\text{weeks}$ deadline.

  $\begin{array}{c}z=\frac{30-32}{\sqrt{1.69}}\\ =\frac{-2}{1.3}\\ =-1.539\end{array}$

Refer probability distribution table.

For $z\le -1.539$ .

  $P\left(z\le -1.539\right)=0.06178$ .

Hence, the probability that the project is completed within $30\text{weeks}$ is $0.06178$ .

(c)

Summary Introduction

Interpretation: The probability that the project is completed within $35\text{weeks}$ is to be determined.

Concept Introduction:

Project Scheduling is used in Project planning and coordination by using Program Evaluation Review Technique (PERT) and Critical Path Method (CPM). PERT is a network based scheduling procedure. This technique is used to plan, schedule and control different project such as construction works, design installation of system, maintenance work and research and development process.

Explanation of Solution

The activity with optimistic, most probable and pessimistic time is tabulated below.

Activity	Optimistic time	Most Probable Time	Pessimistic Time
A	$4$	$5$	$6$
B	$2.5$	$3$	$3.5$
C	$6$	$7$	$8$
D	$5$	$5.5$	$9$
E	$6$	$7$	$8$
F	$1$	$3$	$5$
G	$8$	$10$	$12$

Table (1)

The expression for the estimated time is given by,

  $t_e=\frac{t_0+4t_m+t_p}{6}$

Here, $t_e$ is the estimated time of an activity, $t_0$ is the optimistic time, $t_m$ is the most probable time and $t_p$ is the pessimistic time.

The expression for variance is given by,

  $\text{Variance}=\frac{{\left(t_p-t_0\right)}^2}{36}$

The variance and estimated time of each activity is tabulated below.

Activity	Optimistic time	Most Probable Time	Pessimistic Time	Estimated Time	Variance
A	$4$	$5$	$6$	$5$	$0.111$
B	$2.5$	$3$	$3.5$	$3$	$0.028$
C	$6$	$7$	$8$	$7$	$0.111$
D	$5$	$5.5$	$9$	$6$	$0.444$
E	$6$	$7$	$8$	$7$	$0.111$
F	$1$	$3$	$5$	$3$	$0.444$
G	$8$	$10$	$12$	$10$	$0.444$

Table (2)

Calculate $z$ value to find the probability of meeting a $35\text{weeks}$ deadline.

  $\begin{array}{c}z=\frac{35-32}{\sqrt{1.69}}\\ =\frac{3}{1.3}\\ =2.308\end{array}$

Refer probability distribution table.

For $z\le 2.308$ .

  $P\left(z\le 2.308\right)=0.99134$ .

Hence, the probability that the project is completed within $35\text{weeks}$ is $0.99134$ .

(d)

Summary Introduction

Interpretation: The probability that the project is completed within $32\text{weeks}$ is to be determined.

Concept Introduction:

Project Scheduling is used in Project planning and coordination by using Program Evaluation Review Technique (PERT) and Critical Path Method (CPM). PERT is a network based scheduling procedure. This technique is used to plan, schedule and control different project such as construction works, design installation of system, maintenance work and research and development process.

Explanation of Solution

The activity with optimistic, most probable and pessimistic time is tabulated below.

Activity	Optimistic time	Most Probable Time	Pessimistic Time
A	$4$	$5$	$6$
B	$2.5$	$3$	$3.5$
C	$6$	$7$	$8$
D	$5$	$5.5$	$9$
E	$6$	$7$	$8$
F	$1$	$3$	$5$
G	$8$	$10$	$12$

Table (1)

The expression for the estimated time is given by,

  $t_e=\frac{t_0+4t_m+t_p}{6}$

Here, $t_e$ is the estimated time of an activity, $t_0$ is the optimistic time, $t_m$ is the most probable time and $t_p$ is the pessimistic time.

The expression for variance is given by,

  $\text{Variance}=\frac{{\left(t_p-t_0\right)}^2}{36}$

The variance and estimated time of each activity is tabulated below.

Activity	Optimistic time	Most Probable Time	Pessimistic Time	Estimated Time	Variance
A	$4$	$5$	$6$	$5$	$0.111$
B	$2.5$	$3$	$3.5$	$3$	$0.028$
C	$6$	$7$	$8$	$7$	$0.111$
D	$5$	$5.5$	$9$	$6$	$0.444$
E	$6$	$7$	$8$	$7$	$0.111$
F	$1$	$3$	$5$	$3$	$0.444$
G	$8$	$10$	$12$	$10$	$0.444$

Table (2)

Calculate $z$ value to find the probability of meeting a $32\text{weeks}$ deadline.

  $\begin{array}{c}z=\frac{32-32}{\sqrt{1.69}}\\ =\frac{0}{1.3}\\ =0\end{array}$

Refer probability distribution table.

There is no value for $z=0$ .

Hence, the probability that the project is completed within $32\text{weeks}$ cannot be determined.