Chapter 18, Problem 12PA

Summary Introduction

Interpretation: The total project completion time is to be calculated along with the lowest cost solution if the project is to be completed $3$ weeks early.

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 their immediate predecessor and estimated time, crash time, crash cost and normal cost is tabulated below.

Activity	Immediate Predecessor	Estimated Normal Time	Crash Time	Normal Cost	Crash Cost
A	-	$5$	$4$	$400$	$750$
B	A	$8$	$6$	$1800$	$2200$
C	A	$7$	$6$	$800$	$1100$
D	-	$6$	$5$	$600$	$1000$
E	B, D, C	$8$	$6$	$1700$	$2200$
F	D	$3$	$2$	$800$	$1000$
G	D	$3$	$2$	$500$	$650$
H	E	$4$	$3$	$400$	$600$
I	F, G, H	$6$	$5$	$900$	$1300$

Table (1)

The network diagram for the activity shown in table (1) is shown below.

  

Figure (1)

Therefore, the network diagram of the given project is shown in Figure (1).

The expression for the cost slope is given by,

  $\text{Cost Slope}=\frac{\text{Crash Cost}-\text{Normal Cost}}{\text{Normal Time}-\text{Normal Cost}}$

The cost slope for each activity is tabulated below.

Activity	Immediate Predecessor	Estimated Normal Time	Crash Time	Normal Cost	Crash Cost	Cost Slope
A	-	$5$	$4$	$400$	$750$	$350$
B	A	$8$	$6$	$1800$	$2200$	$200$
C	A	$7$	$6$	$800$	$1100$	$300$
D	-	$6$	$5$	$600$	$1000$	$400$
E	B, D, C	$8$	$6$	$1700$	$2200$	$250$
F	D	$3$	$2$	$800$	$1000$	$200$
G	D	$3$	$2$	$500$	$650$	$150$
H	E	$4$	$3$	$400$	$600$	$200$
I	F, G, H	$6$	$5$	$900$	$1300$	$400$

Table (2)

Consider figure (1), the different paths and their completion times is tabulated below.

Path	Completion time (weeks)
$\text{A-B-E-H-I}$	$5+8+8+4+6=31$
$\text{A-C-E-H-I}$	$5+7+8+4+6=30$
$\text{D-E-H-I}$	$6+8+4+6=24$
$\text{D-G-I}$	$6+3+6=15$
$\text{D-F-I}$	$6+3+6=15$

Table (3)

From Table (3), the maximum completion time is taken by path $\text{A-B-E-H-I}$ . So, this is the critical path.

Therefore, the critical path is $\text{A-B-E-H-I}$ and its duration is $31\text{weeks}$ .

Now the project is to be completed $3$ weeks earlier.

Consider the critical path activities.

The lowest crash cost per week is of activity B and activity H.

Crash Activity B by $2$ weeks and activity H by $\text{1}\text{week}$ .

  $\begin{array}{c}\text{Crash B}=2\times 200\\ =400\end{array}$

And,

  $\begin{array}{c}\text{Crash H}=1\times 200\\ =200\end{array}$

Thus, the additional cost of crashing the activity by $3\text{weeks}$ is,

  $\begin{array}{c}\text{Additional cost}=200+400\\ =600\end{array}$

The new network diagram is shown in figure below.

  

Figure (2)

Consider figure (2), the different paths and their completion times is tabulated below.

Path	Completion time (weeks)
$\text{A-B-E-H-I}$	$5+7+7+3+6=28$
$\text{A-C-E-H-I}$	$5+7+8+3+6=29$
$\text{D-E-H-I}$	$6+8+3+6=23$
$\text{D-G-I}$	$6+3+6=15$
$\text{D-F-I}$	$6+3+6=15$

Table (4)

From Table (4), the maximum completion time is taken by path $\text{A-C-E-H-I}$ . So, this is the critical path.

Therefore, the critical path is $\text{A-C-E-H-I}$ and its duration is $29\text{weeks}$ .

But the desired project duration is $28\text{ weeks}$ .

Try different option.

Crash Activity B by $1$ week, activity E by $1\text{week}$ and activity H by $\text{1}\text{week}$ .

  $\begin{array}{c}\text{Crash B}=1\times 200\\ =200\end{array}$

And,

  $\begin{array}{c}\text{Crash H}=1\times 200\\ =200\end{array}$

And,

  $\begin{array}{c}\text{Crash E}=1\times 250\\ =250\end{array}$

Thus, the additional cost of crashing the activity by $3\text{weeks}$ is,

  $\begin{array}{c}\text{Additional cost}=200+200+250\\ =650\end{array}$

The new network diagram is shown in figure below.

  

Figure (3)

Consider figure (3), the different paths and their completion times is tabulated below.

Path	Completion time (weeks)
$\text{A-B-E-H-I}$	$5+7+7+3+6=28$
$\text{A-C-E-H-I}$	$5+7+7+3+6=28$
$\text{D-E-H-I}$	$6+7+3+6=22$
$\text{D-G-I}$	$6+3+6=15$
$\text{D-F-I}$	$6+3+6=15$

Table (4)

From Table (4), the maximum completion time is taken by path $\text{A-C-E-H-I}$ and $\text{A-B-E-H-I}$ So, this are the critical path.

Therefore, the critical path are $\text{A-C-E-H-I and A-B-E-H-I}$ and its duration is $28\text{weeks}$ .

Therefore, the crashing cost of the project is $\$650$ .