# PERT and CPM

 Question 1
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 1 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 2
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 2 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 3
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 3 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
 Question 4
The activities of a project, their duration and the precedence relationships are given in the table. For example, in a precedence relationship $"X \lt Y \lt Z"$ means that X is predecessor of activities Y and Z. The time to complete the activities along the critical path is ______ weeks.
 A 17 B 21 C 23 D 25
GATE ME 2019 SET-2   Industrial Engineering
Question 4 Explanation:

Critical path duration = 23 weeks
 Question 5
The arc lengths of a directed graph of a project are as shown in the figure. The shortest path length from node 1 to node 6 is _______.
 A 7 B 9 C 11 D 14
GATE ME 2018 SET-2   Industrial Engineering
Question 5 Explanation:
Shortest path is

Shortest length = 7
It is the problem of shortest path which will be 7.
In this question do not confuse with critical path. Examinar ask shortest path. Critical path is the longest path which is not asked in this question.
 Question 6
Processing times (including setup times) and due dates for six jobs waiting to be processed at a work centre are given in the table. The average tardiness (in days) using shortest processing time rule is ___________ (correct to two decimal places).
 A 4.26 B 5.65 C 6.33 D 8.46
GATE ME 2018 SET-1   Industrial Engineering
Question 6 Explanation:
By SPT Rule

Total tardiness = 38
Average tardiness per job $=\frac{\text { Total tardiness }}{\text { No. of Jobs }}=\frac{38}{6}=6.33\text{days}$
 Question 7
A project starts with activity A and ends with activity F. The precedence relation and durations of the activities are as per the following table:

The minimum project completion time (in days) is _______.
 A 19 B 30 C 39 D 48
GATE ME 2017 SET-2   Industrial Engineering
Question 7 Explanation:

There are two path,
\begin{aligned} A-B-D-F=30 \text { days } \\ A-C-E-F=24 \text { days } \end{aligned}
Critical path A-B-D-F and of 30 days and it is also minimum time required to complete project.
 Question 8
The standard deviation of linear dimensions P and Q are 3 m and 4 m, respectively. When assembled, the standard deviation (in m) of the resulting linear dimension (P+Q) is ________
 A 5 B 6 C 4 D 3
GATE ME 2017 SET-2   Industrial Engineering
Question 8 Explanation:
According to Root Sum Square or RSS model
$\sigma_{a}=\sqrt{\sum_{i=1}^{n} \sigma_{i}^{2}}$
where, $\sigma_{a}=$ assembly tolerance standard deviation
$\sigma_{i}=i^{\text {th }}$ component tolerance standard deviation
Here $\sigma_{P}=3 \mu \mathrm{m}$ and $\sigma_{Q}=4 \mu \mathrm{m}$
$\sigma_{a}=\sqrt{\sigma_{P}^{2}+\sigma_{Q}^{2}}=\sqrt{3^{2}+4^{2}}=5 \mu \mathrm{m}$
 Question 9
A sample of 15 data is follows: 17, 18, 17, 17, 13, 18, 5, 5, 6, 7, 8, 9, 20, 17, 3. The mode of the data is
 A 4 B 13 C 17 D 20
GATE ME 2017 SET-2   Industrial Engineering
Question 9 Explanation:
Mode means highest number of observations or occurrence of data most of the time as data 17, occurs four times, i.e., highest time. So mode is 17.
 Question 10
In PERT chart, the activity time distribution is
 A Normal B Binomial C Poisson D Beta
GATE ME 2016 SET-3   Industrial Engineering
Question 10 Explanation:
Activity time distribution is beta.
There are 10 questions to complete.