# PERT and CPM

 Question 1
A project consists of five activities (A, B, C, D and E). The duration of each activity follows beta distribution. The three time estimates (in weeks) of each activity and immediate predecessor(s) are listed in the table. The expected time of the project completion is __________ weeks (in integer).
$\begin{array}{|c|c|c|c|c|} \hline \text{Activity} & \text{Optimistic time (week)} & \text{Most likely time (week)} &\text{Pessimistic time (week)} &\text{Immediatepredecessor(s)} \\ \hline A& 4 & 5 &6 & None\\ \hline B& 1& 3 & 5 & A\\ \hline C& 1& 2 & 3 & A\\ \hline D& 2& 4 & 6 & C\\ \hline E& 3& 4 & 5 &B,D \\ \hline \end{array}$
 A 12 B 15 C 16 D 18
GATE ME 2022 SET-2   Industrial Engineering
Question 1 Explanation:
$\begin{array}{|c|c|} \hline \text{Activity}& t_e=\frac{t_0+4t_m+t_p}{6}\\ \hline A&5\\ \hline B&3\\ \hline C&2\\ \hline D&4\\ \hline E&4 \\ \hline \end{array}$ $\begin{array}{|c|c|} \hline \text{Path}& \text{Duration}\\ \hline 1-2-4-5(A-B-E)&12\\ \hline 1-2-3-4-5 (A-C-D-E)&15\\ \hline \end{array}$
Expected completion time of the project = critical path duration = 15 weeks
 Question 2
Activities A to K are required to complete a project. The time estimates and the immediate predecessors of these activities are given in the table. If the project is to be completed in the minimum possible time, the latest finish time for the activity G is _____ hours.
$\begin{array}{|c|c|c|}\hline \text{Activity} &\text{Time (hours)} & \text{Immediate predecessors}\\ \hline A &2 &- \\ \hline B& 3 &- \\ \hline C& 2 &- \\ \hline D & 4 &A \\ \hline E & 5 & B\\ \hline F & 4 & B\\ \hline G & 3 & C\\ \hline H & 10 &D,E \\ \hline I & 5 & F\\ \hline J & 8 & G\\ \hline K & 3 &H,I,J\\ \hline \end{array}$
 A 5 B 10 C 8 D 9
GATE ME 2022 SET-1   Industrial Engineering
Question 2 Explanation: Activity 'G' LFT of activity 'G' = 10 hours

 Question 3
A PERT network has 9 activities on its critical path. The standard deviation of each activity on the critical path is 3. The standard deviation of the critical path is
 A 3 B 9 C 27 D 81
GATE ME 2021 SET-2   Industrial Engineering
Question 3 Explanation:
In CPM,
$\begin{array}{l} \sigma=\sqrt{\text { sum of variance along critical path }} \\ \sigma=\sqrt{\sigma^{2}+\sigma^{2}+\ldots .+\sigma^{2}} \\ \sigma=\sqrt{9 \sigma^{2}}=\sqrt{9 \times 9}=9 \end{array}$
 Question 4
Activities A, B, C and D form the critical path for a project with a PERT network. The means and variances of the activity duration for each activity are given below. All activity durations follow the Gaussian (normal) distribution, and are independent of each other.

$\begin{array}{|l|c|c|c|c|}\hline \text{Activity} & \text{A} & \text{B} & \text{C} & \text{D} \\ \hline \text{Mean (days)} & \text{6} & \text{11} & \text{8} & \text{15} \\ \hline \text{Variance (days^{2})} & \text{4} & \text{9} & \text{4} & \text{9}\\ \hline \end{array}$

The probability that the project will be completed within 40 days is _____ (round off to two decimal places).
(Note: Probability is a number between 0 and 1).
 A 0.25 B 0.5 C 0.65 D 0.85
GATE ME 2021 SET-1   Industrial Engineering
Question 4 Explanation:

PERT-CPM
\begin{aligned} T_{S} &=40 \text { days, } T_{E}=6+11+8+15 \\ T_{E} &=40 \text { days, } \\ 2 &=\frac{T_{S}-T_{E}}{\sigma}=0 \rightarrow 50 \% \end{aligned} Probability of completing project in expected time is always 0.5.
 Question 5
Consider the following network of activities, with each activity named A-L, illustrated in the nodes of the network The number of hours required for each activity is shown alongside the nodes. The slack on the activity L, is ________ hours.
 A 1 B 2 C 3 D 4
GATE ME 2020 SET-2   Industrial Engineering
Question 5 Explanation:
Time along path A-B-C-F-I-J-K = 42 hours
Time along path A-B-C-E-H-L = 31 hours
Time along path A-B-C-D-G-H-L = 40 hours
Slack for L = 42 - 40 = 2 hours

There are 5 questions to complete.