Pessimistic time and optimistic time of completion of an activity are given as 10 days and 4 days respectively, the variance of the activity will be
Concept:
Project Evaluation and Review Technique (PERT) is probabilistic in nature and is based upon threetime estimates to complete an activity.
Optimistic Time (to): It is the minimum time that will be taken to complete an activity if everything goes according to the plan.
Pessimistic Time (tp): It is the maximum time that will be taken to complete an activity when everything goes against the plan.
Most likely time (tm): It is the time required to complete a project when an activity is executed under normal conditions.
Average or most expected time is given by \({t_E} = \left( {\frac{{{t_p}\; + \;4{t_m}\; + {t_o}}}{6}} \right)\)
The variance gives the measure of uncertainty of activity completion. The variance of the activity is given by
Variance, \(V = {\left( {\frac{{{t_p}  {t_0}}}{6}} \right)^2}\)
Standard duration, \(\sigma = \sqrt {variance} \)
Calculation:
Given:
t_{p} = 10 days, t_{o} = 4 days
\({\rm{V}} = {\left( {\frac{{{{\rm{t}}_{\rm{p}}}  {{\rm{t}}_{\rm{o}}}}}{6}} \right)^2} = {\left( {\frac{{10  4}}{6}} \right)^2} = 1\)
The variance of the activity is 1.
Concept:
In a transportation problem with m supply points and n demand points
Number of constraints = m + n
Number of variables = m × n
Number of equations = m + n  1
Calculation:
Given:
m = 4, n = 5
Number of constraints = m + n = 4 + 5 = 9
Explanation:
CPM does not directly model uncertainty.
PERT was developed to address the needs of projects which are being done for the first time – a challenge to estimate activity duration.
PERT (Program Evaluation and Review Technique) uses 3 cases:
PERT determines the probability for each duration, whereas CPM considers the most likely duration.
In the standard PERT analysis, the distribution assumed for the activity times is a Beta distribution.
Concept:
Total float: It is the amount of time that the completion time of an activity can be delayed without affecting project completion time.
Free float: It is the amount of time that the activity completion time can be delayed without affecting the earliest start time of the immediate successor activities in the network.
Interfering Float: Maximum amount by which an activity can be delayed without delaying the project but will cause delay to the Early Start of some following activity
Independent Float: Amount by which an activity can be delayed without delaying the project; even if all predecessors are at Late Finish and all Successors are at Early Start.Explanation:
Critical Path:
A critical path is a sequence of interdependent activities or tasks that must be finished before the project can be finished. It is the longest path (i.e. path with the longest duration) from project start to finish.
Critical Path Method:
For the following LPP 
Max. Z = 0.1 x_{1} + 0.5 x_{2}
2x_{1} + 5x_{2} ≤ 80
x_{1} + x_{2} ≤ 20
x_{1}, x_{2} ≥ 0
to get the optimum solution, the values of x_{1}, x_{2} are 
Concept:
Convert the inequality constraints into equations and find the common points of the bounded region.
Calculations:
Given, LPP:
Max. Z = 0.1 x_{1} + 0.5 x_{2}
2x_{1} + 5x_{2} ≤ 80
x_{1} + x_{2} ≤ 20
x_{1}, x_{2} ≥ 0
Convert the inequality constraints into equations, we have
2x_{1} + 5x_{2} = 80
x_{1} + x_{2} = 20
2x_{1} + 5x_{2} = 80 passes through the point (0, 16) and (40, 0)
x_{1} + x_{2} = 20 passes through the point (0, 20) and (20, 0).
Now, the co ordinates of the point A = (0, 16), B = (20, 0) and C = \(\left(\dfrac{20}{3}, \dfrac{40}{3}\right)\)
Corner Point  Coordinate of point  value of Z 
A  (0, 16)  8 
B  (20, 0)  2 
C  \(\left(\dfrac{20}{3}, \dfrac{40}{3}\right)\)  6.066 
Here, maximum value occurs at (0, 16)
For given
Max. Z = 0.1 x_{1} + 0.5 x_{2}
2x_{1} + 5x_{2} ≤ 80
x_{1} + x_{2} ≤ 20
x_{1}, x_{2} ≥ 0
to get the optimum solution, the values of x_{1}, x_{2} are (0, 16)
Explanation
Slack or Event Float
Slack
There are three types of floats.
Total Float (TF) 

Free Float (FF) 
· Part of the Total Float, which can be used without affecting the float of succeeding activity. · Extra time by which an activity can be delayed so that the succeeding activity can be started on earliest start time.

Independent Float (IF) 

Network models can be used as an aid in scheduling large complex projects that consist of many activities.
PERT
If the duration of the activities is not known with certainty, the Program Evaluation and Review Technique (PERT) can be used to estimate the probability that the project will be completed by a given deadline.
PERT approach takes account of the uncertainties. In this approach, 3time values are associated which each activity. So it is probabilistic.
CPM
If the duration of each activity is known with certainty, then the critical path method (CPM) can be used to determine the length of time required to complete a project.
Whereas CPM involves the critical path which is the largest path in the network from starting to ending event and defines the minimum time required to complete the project. So it is deterministic.
CPM 
PERT 
CPM is an activityoriented network diagram 
PERT is an eventoriented network diagram 
CPM is based upon a deterministic approach 
It is based on Probabilistic approach 
Only one time estimates are made for each activity 
Threetime estimates are made for each activity 
Each activity follows a normal distribution 
Each activity follows β distribution 
Concept:
No. of customers in the system, \({L_S} = \frac{\lambda }{{\mu  \lambda }}\)
No. of customers in the queue, \({L_q} = \frac{{{\lambda ^2}}}{{\mu \left( {\mu  \lambda } \right)}}\)
Calculation:
Arrival rate λ = 10 minute/customer = 6 customers/hour
Service rate, μ = 6 minute/customer = 10 customers/hour
The average length of the queue, \({L_q} = \frac{{{\lambda ^2}}}{{\mu \left( {\mu  \lambda } \right)}}\)
\(\;{L_q} = \frac{{{6^2}}}{{10\;\left( {10\;  \;6} \right)}} = \;0.9 = \frac{9}{{10}}\)
Points to remember
Concept
Free float is the portion of total float by which an activity can be delayed without affecting succeeding activity.
\({F_F} = \left( {T_E^j  T_E^i} \right)  tij\)
Free float is also equal to total float minus head event slack.
\({F_F} = {F_T}  S_j\)
which implies that free float do not affect succeeding activity but affect preceeding activities.
Total Float: It is the maximum delay possible for an activity without considering any delay in its precedence or succeeding activity.
Independent float is the maximum delay possible for an activity which used floats of preceding activity, but we will not affect the float of succeeding activity.
Following data refers to the activities of a project, where, node 1 refers to the start and node 5 refers to the end of the project
Activity 
Duration (days) 
12 
2 
23 
1 
43 
3 
14 
3 
25 
3 
35 
2 
45 
4 
The critical path (CP) in the network is
Concept:
Trick: Whenever activity is mentioned take points on node itself to avoid the confusion.
And take the duration on path.
→ Note: Find the durations of path given in option and then take the path with max duration
Because, Critical path is the minimum time required to completed the task or path with maximum duration
1) 1 – 2 – 3 – 5 = 5 days
2) 1 – 2 – 3 – 4 – 5 (No path is formed) ∵ 3 – 4 path is not a path
3) 1 – 4 – 3 – 5 = 8 days
4) 1 – 4 – 5 → 7 days
Hence from the above calculation we can see that path in the option(c) is the critical path.
Explanation:
Critical activity:
Critical activities are those activities whose float is zero and they lie on the critical path.
Float:
Relax or delay provided to any activity is known as float.
Floats are of three types. They are as follows:
Total Float (TF):
The maximum delay or relax provided to any activity without affecting the project duration.
TF = L_{j} – E_{i} – d_{ij}
Free Float (FF):
Relax or delay provided to any activity without affecting the Earliest Start Time (EST) of the successor activity.
FF = E_{j} – E_{i} – d_{ij}
Independent Float:
Relax or delay provided to any activity without affecting the Earliest Start Time (EST) of successor activity as well as Latest Finish Time (LFT) of the predecessor activity.
IF = E_{j} – L_{i}  d_{ij}
Relation among float TF ≥ FF ≥ IF
Critical Path:
Explanation:
Crashing is the method for shortening the project duration by reducing the time of one or more critical activities to less than their normal time. In crashing if cost increases then time decreases.
Steps involved in crashing a project Network.
Explanation
Duality
Primal (Maximisation) 
Dual ( Minimisation) 
i^{th }constraint ≤ 
i^{th} variable ≥ 0 
i^{th} constraint ≥ 
i^{th} variable ≤ 0 
i^{th} constraint = 
j^{th }variable unrestricted 
j^{th} variable ≥ 0 
j^{th }constraint ≥ 
j^{th} variable ≤ 0 
j^{th} constraint ≤ 
j^{th} variable unrestricted 
j^{th} constraint = 
A PERT activity has an optimistic time of 3 days, pessimistic time of 15 days and an expected time of 7 days. The most likely time of the activity is
Concept:
Expected time in a PERT activity is given by:
\({T_e} = \frac{{{t_o} + {t_p} + 4{t_n}}}{6}\)
where,
t_{o} = Optimistic time, t_{p} = Pessimistic time, t_{m} = Most likely time
Calculation:
Given:
T_{e }= 7 days, t_{o} = 3 days, t_{p} = 15 days
\({T_e} = \frac{{{t_o} + {t_p} + 4{t_m}}}{6}\; \Rightarrow 7 = \frac{{15 + 3 + 4{{t_m}}}}{6} \)
t_{m} = 6 days
Explanation:
Crashing is the method for shortening the project duration by reducing the time of one or more critical activities to less than their normal time. In crashing if cost increases then time decreases.
A project has four activities P, Q, R and S as shown below.
Activity 
Normal duration (days) 
Predecessor 
Cost slope (Rs./day) 
P 
3 
 
500 
Q 
7 
P 
100 
R 
4 
P 
400 
S 
5 
R 
200 
The normal cost of the project is Rs. 10,000/ and the overhead cost is Rs. 200/ per day. If the project duration has to be crashed down to 9 days, the total cost (in Rupees) of the project is _______
Concept:
Crashing:
\(Cost\;slope = \frac{{Crashed\;cost  Normal\;cost}}{{Normal\;time  Crash\;time}}\)
Calculation:
Given:
The normal cost of the project = Rs 10,000
Overhead cost = Rs 200/day
Various paths possible are:
P – R – S = 12 days (C.P)
P – Q = 10 days
Direct cost for 12 days = Rs 10000
Indirect cost for 12 days = 200 × 12 = Rs 2400
Total cost for 12 days = direct cost + indirect cost
= 10000 + 2400 = Rs 12400
Now, the next critical path is 10 days.
Activity 
Possible crash 
Cost slope (Rs/day) 
P 
3 
500 
R 
4 
400 
S 
5 
200 (minimum) 
Crash S by 2 days = 2 × 200 = Rs 400
(Total cost)_{10 days} = direct cost + indirect cost
⇒ (Total cost)_{10 days} = (10000 + 400) + (200 × 10)
⇒ (Total cost)_{10 days} = Rs 12,400
Now we have to crash the project to 9 days
Activity 
Possible crash 
Cost slope (Rs/day) 
P 
3 
500 
R 
4 
400 
S 
3 
200 (minimum) 
Q 
7 
100 (minimum) 
Crash S and Q by 1 day = 200 + 100 = Rs 300
(Total cost)_{9 days} = direct cost + indirect cost
⇒ (Total cost)_{9 days} = (10400 + 300) + (200 × 9)
⇒ (Total cost)_{9}_{ days} = Rs 12,500
Cars arrive at a service station according to Poisson’s distribution with a mean rate of 5 per hour. The service time per car is exponential with a mean of 10 minutes. At steady state, the average waiting time in the queue is
Concept:
Waiting time in the queue \(W_q=\frac{L_q}{λ}\) and \(L_q=\frac{ρ^2}{1ρ}\)
\(\rho=\frac{\lambda}{μ}\)
where, L_{q} = length of queue, λ = arrival rate, μ = service rate
Calculation:
Given:
λ = 5 cars per hour, μ = 1 car per 10 minute = 6 cars per hour
⇒ \(\rho =\frac{5}{6}\)
\(L_q= \frac{(5/6)^2}{1(5/6)}=\frac{25}{6}\)
\(W_q=\frac{25/6}{5}=\frac{5}{6}\) hours = \(\frac{5}{6}\times60=50~min\)
Concept:
In CPM:
The standard deviation of critical path:
σ_{cp }= \(\sqrt {Sum\;of\;variance\;along\;critical\;path} \)
σ_{cp} = \(\sqrt {σ _1^2 + σ _2^2 + \ldots + σ _8^2 + σ _9^2} \)
Where, σ_{1}, σ_{2}, ...., σ_{8}, σ_{9} are the standard deviation of each activity on the critical path
Calculation:
Given:
σ1, σ2, ...., σ8, σ9 = 3
σcp = \(\sqrt {σ _1^2 + σ _2^2 + \ldots + σ _8^2 + σ _9^2} \)
σ_{cp} = \(\sqrt {3^2 + 3^2 + 3^2 + 3^2 + 3^2 + 3^2 + 3^2 + 3^2 + 3^2} \)
σ_{cp} = \(\sqrt {9 \times 9} \) = 9
∴ the standard deviation of the critical path is 9.
Concept:
There are four types of customer behaviours.