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GNDU QUESTION PAPERS 2024
B.com 6
th
SEMESTER
OPERATIONS RESEARCH
Time Allowed: 3 Hours Maximum Marks: 50
Note: Aempt Five quesons in all, selecng at least One queson from each secon. The
Fih queson may be aempted from any Secon. All quesons carry equal marks.
SECTION—A
1. Explain Operational Research. Discuss its characteristics and limitations.
2. Rewrite the following LPP in standardised form for application of simplex method :
Maximize
Z = 8x₁ − 6x₂ + 7x₃ + 2x₄
subject to
4x₁ + 3x₂ + 6x₃ + x₄ ≤ 40
−x₁ + 2x₂ + 3x₃ + x₄ ≤ 5
9x₁ − 5x₂ + 7x₃ − x₄ ≥ 60
6x₂ + 2x₃ + 4x₄ = 47
x₁, x₂, x₃, x₄ ≥ 0
SECTION—B
3. Solve the following transportation problem for minimum cost :
Destination
A
B
C
D
Requirement
1
7
4
3
4
15
2
3
2
7
5
25
3
4
4
3
7
20
4
9
7
5
3
40
Availabilities: 12, 8, 35, 25
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4. A company solicits bids on each of four projects from five contractors. Only one project
may be assigned to any contractor. The bids received (in thousands of rupees) are given in
the accompanying table. Contractor D feels unable to carry out project 3 and therefore,
submits no bid.
Project
A
B
C
D
E
1
18
25
22
26
25
2
26
29
26
27
24
3
28
31
30
–
31
4
26
28
27
26
29
SECTION—C
5. Explain Game Theory. Discuss the steps of solving Game Theory problem by Dominance
method by giving suitable examples.
6. A post office has two clerks, either of whom averages 1.5 minutes per customer
transaction (the service time being distributed exponentially). The arrival rate of the
customers to the post office is one customer per minute. Compute:
(a) The probability that both the clerks would be idle.
(b) The probability that there shall be one customer in the post office.
(c) The probability that there shall be five customers in the post office.
(d) The average number of customers waiting in the queue.
(e) The average number of customers being served.
(f) The average time a customer spends waiting for service.
(g) The average time a customer spends in a post office.
SECTION—D
7. What do you understand by CPM? State the five steps of the working methodology of
critical path analysis. Can a critical path change during the course of a period?
8. A project consists of eight activities with the following time estimates:
Activity
Immediate Predecessor
Optimistic
Most Likely
Pessimistic
A
–
1
1
7
B
–
1
4
7
C
–
2
2
8
D
A
1
1
1
E
B
2
5
14
F
C
2
5
8
G
D & E
3
6
15
H
F & G
1
2
3
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Required:
(a) Draw PERT network.
(b) Find the expected time for each activity.
(c) Determine the earliest event times and latest allowable times.
(d) Determine the critical path.
(e) Determine the total slack for each activity.
(f) What is the probability that the project will be completed in
(i) 22 days (ii) 18 days (iii) 19 days?
(g) What project duration will have 95% chance of completion?
(h) If the average duration for activity F increases to 14 days,
what will be its effect on the expected project completion time which will have 95%
confidence?
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GNDU ANSWER PAPERS 2024
B.com 6
th
SEMESTER
OPERATIONS RESEARCH
Time Allowed: 3 Hours Maximum Marks: 50
Note: Aempt Five quesons in all, selecng at least One queson from each secon. The
Fih queson may be aempted from any Secon. All quesons carry equal marks.
SECTION—A
1. Explain Operational Research. Discuss its characteristics and limitations.
Ans: What is Operational Research?
Imagine you are running a business, like a car wash shop, and you want to decide:
• How many workers should you hire?
• How much water and soap should you use?
• How can you serve more customers in less time?
These are decision-making problems, and this is exactly where Operational Research (OR)
helps.
Operational Research is a scientific method of solving problems and making better
decisions using mathematics, data, and logical analysis.
In simple words:
OR helps us choose the best solution from many possible options.
Simple Definition
Operational Research is the application of scientific and mathematical techniques to
improve decision-making and efficiency in organizations.
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How Operational Research Works (Process)
Here’s a simple diagram to understand the OR process:
Problem Identification
↓
Data Collection
↓
Model Building (Mathematical Model)
↓
Analysis & Solution
↓
Implementation
↓
Review & Improvement
Think of it like solving a puzzle step by step:
• First, understand the problem
• Then collect facts
• Build a solution
• Apply it and improve
Example to Understand OR Easily
Suppose a delivery company wants to minimize fuel cost.
• OR will help find the shortest route
• It will calculate time, distance, traffic
• Then suggest the best possible path
This is called optimization—a key concept in OR.
Characteristics of Operational Research
Now let’s look at the important features of OR:
1. Scientific Approach
OR follows a step-by-step scientific method:
• Observation
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• Analysis
• Testing
It is not based on guesswork but on logic and data.
2. Decision-Oriented
The main goal of OR is to help managers make better decisions.
Example: Should a company increase production or not?
3. Use of Mathematical Models
OR uses equations and formulas to represent real-life problems.
Example:
• Profit = Revenue – Cost
• Optimization models
4. Interdisciplinary Nature
OR combines knowledge from different fields:
• Mathematics
• Economics
• Statistics
• Computer Science
It is like teamwork of different subjects.
5. Focus on Optimization
OR always tries to:
• Maximize profit
• Minimize cost
• Improve efficiency
It finds the best solution, not just any solution.
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6. Use of Computers
Modern OR depends heavily on computers for:
• Data analysis
• Complex calculations
Without computers, OR would be very slow.
7. System Approach
OR studies the whole system, not just one part.
Example:
Instead of improving only production, it looks at:
• Production
• Supply
• Distribution
8. Quantitative Analysis
OR focuses on numbers and measurable data.
It avoids emotional or personal decisions.
Limitations of Operational Research
Even though OR is very powerful, it has some limitations:
1. Depends on Accurate Data
If the data is wrong, the results will also be wrong.
“Garbage in, garbage out.”
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2. Ignores Human Factors
OR mainly focuses on numbers and may ignore:
• Human emotions
• Behavior
• Motivation
But real-life decisions involve people too.
3. Complex and Costly
OR techniques can be:
• Difficult to understand
• Expensive to implement
Small businesses may not afford it.
4. Requires Skilled Experts
OR needs trained professionals like:
• Analysts
• Mathematicians
Not everyone can use OR easily.
5. Time-Consuming
Building models and analyzing data can take a lot of time.
Not suitable for urgent decisions.
6. Assumptions May Be Unrealistic
OR models are based on assumptions, which may not match real life.
Example:
Assuming demand is constant—but in reality, it changes.
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7. Not a Complete Solution
OR only assists decision-making, it does not replace human judgment.
Final decisions are still taken by managers.
Conclusion
Operational Research is like a smart assistant that helps businesses and organizations make
better decisions using data and logic.
• It improves efficiency
• Saves time and cost
• Provides scientific solutions
But at the same time:
• It cannot replace human thinking
• It depends on correct data
• It can be complex and costly
So, the best approach is to combine OR with human experience and judgment.
2. Rewrite the following LPP in standardised form for application of simplex method :
Maximize
Z = 8x₁ − 6x₂ + 7x₃ + 2x₄
subject to
4x₁ + 3x₂ + 6x₃ + x₄ ≤ 40
−x₁ + 2x₂ + 3x₃ + x₄ ≤ 5
9x₁ − 5x₂ + 7x₃ − x₄ ≥ 60
6x₂ + 2x₃ + 4x₄ = 47
x₁, x₂, x₃, x₄ ≥ 0
Ans: Step 1: Understanding the Problem
We are asked to maximize:
subject to the following constraints:
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1.
2.
3.
4.
5.
Step 2: What Does “Standard Form” Mean?
For the simplex method, the LPP must be written in a standardized form. That means:
1. The objective function should be in terms of maximization (already given here).
2. All constraints should be written as equalities (not inequalities).
3. All variables must be non-negative.
4. If there are “≥” constraints, we need to introduce surplus variables and possibly
artificial variables.
5. If there are “≤” constraints, we add slack variables to convert them into equalities.
6. If there are “=” constraints, sometimes artificial variables are added to start the
simplex method.
So, the job is to transform each inequality/equality into a neat equation with extra variables.
Step 3: Rewriting Each Constraint
Constraint 1:
For “≤” constraints, we add a slack variable
:
Constraint 2:
Again, add a slack variable
:
Constraint 3:
For “≥” constraints, we subtract a surplus variable and add an artificial variable (to start the
simplex method). Let’s call them
(surplus) and
(artificial):
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where
.
Constraint 4:
This is already an equality. But since it’s not a “≤” type, we add an artificial variable
:
Step 4: Objective Function in Standard Form
The objective function is:
In simplex method, we usually write it as:
This way, it looks like an equation.
Step 5: Final Standardized Form
Now, putting it all together:
Maximize:
Subject to:
with
Why Do We Do This?
Think of slack, surplus, and artificial variables as “adjustments” that help balance the
equations:
• Slack variables fill the gap in “≤” inequalities.
• Surplus variables cut down the excess in “≥” inequalities.
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• Artificial variables are temporary helpers to start the simplex method when
equations don’t naturally fit.
Without these, the simplex method cannot begin because it requires a neat system of
equations.
Final Thought
So, rewriting an LPP in standard form is like preparing the ground before planting seeds.
You’re not solving yet—you’re just making sure the soil (equations) is neat, balanced, and
ready. Once everything is in standard form, the simplex method can be applied step by step
to find the optimal solution.
SECTION—B
3. Solve the following transportation problem for minimum cost :
Destination
A
B
C
D
Requirement
1
7
4
3
4
15
2
3
2
7
5
25
3
4
4
3
7
20
4
9
7
5
3
40
Availabilities: 12, 8, 35, 25
Ans: Understanding the Problem (In Simple Words)
Imagine you are managing a transportation system where goods must be sent from 4
sources (rows) to 4 destinations (columns A, B, C, D).
Each route has a cost, and your goal is:
Send goods in such a way that all requirements are satisfied at minimum total cost.
Given Data
Cost Table
Source → / Destination ↓
A
B
C
D
Supply
1
7
4
3
4
15
2
3
2
7
5
25
3
4
4
3
7
20
4
9
7
5
3
40
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Requirements (Demand)
Destination
A
B
C
D
Demand
12
8
35
25
Step 1: Check Balance
Total Supply = 15 + 25 + 20 + 40 = 100
Total Demand = 12 + 8 + 35 + 25 = 80
Since Supply > Demand, this is an unbalanced problem.
Solution:
Add a dummy destination (E) with demand = 20 (100 − 80)
All costs to dummy = 0.
Step 2: Use Vogel’s Approximation Method (VAM)
This method helps us find a good starting solution by choosing the least-cost routes
smartly.
Iteration 1
Calculate penalties (difference between smallest & second smallest cost in rows/columns).
• Row 4 has highest penalty → choose row 4
• Minimum cost in row 4 = 3 (Destination D)
Allocate:
Min(40, 25) = 25 units to (4, D)
Update:
• Supply row 4 → 15 left
• Demand D → 0 (fulfilled)
Iteration 2
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Recalculate penalties.
• Column B has highest penalty
• Minimum cost in column B = 2 (Source 2)
Allocate:
Min(25, 8) = 8 units to (2, B)
Update:
• Supply row 2 → 17 left
• Demand B → 0
Iteration 3
• Column A chosen
• Minimum cost = 3 (Source 2)
Allocate:
Min(17, 12) = 12 units to (2, A)
Update:
• Supply row 2 → 5 left
• Demand A → 0
Iteration 4
Only column C left (demand 35)
Allocate from cheapest routes:
• (1, C) = 3 → allocate 15
• (3, C) = 3 → allocate 20
Now demand C = 0
Remaining
Remaining supply:
• Row 2 → 5
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• Row 4 → 15
These go to dummy column (cost = 0)
Final Allocation Table
Source → / Destination ↓
A
B
C
D
Dummy
1
0
0
15
0
0
2
12
8
0
0
5
3
0
0
20
0
0
4
0
0
0
25
15
Step 3: Calculate Total Cost
Now multiply allocation × cost:
• (1, C) → 15 × 3 = 45
• (2, A) → 12 × 3 = 36
• (2, B) → 8 × 2 = 16
• (3, C) → 20 × 3 = 60
• (4, D) → 25 × 3 = 75
• Dummy allocations → 0 cost
Total Minimum Cost
Total = 45 + 36 + 16 + 60 + 75 = 232
Final Answer
Minimum Transportation Cost = 232
Concept Made Easy (Quick Recap)
Think of this problem like:
• You have warehouses (sources) and shops (destinations)
• Each route costs money
• You want to deliver all goods at lowest cost
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Key Steps:
1. Balance the problem (add dummy if needed)
2. Use VAM to get initial solution
3. Allocate where cost is lowest and penalty is highest
4. Calculate total cost
Simple Visual Diagram
Sources → A B C D
--------------------------
1 (15) | | | 15 | |
2 (25) | 12 | 8 | | |
3 (20) | | | 20 | |
4 (40) | | | | 25 |
--------------------------
Final Thought
Transportation problems may look complex at first, but once you understand the logic of
allocation and cost minimization, they become almost like solving a puzzle.
4. A company solicits bids on each of four projects from five contractors. Only one project
may be assigned to any contractor. The bids received (in thousands of rupees) are given in
the accompanying table. Contractor D feels unable to carry out project 3 and therefore,
submits no bid.
Project
A
B
C
D
E
1
18
25
22
26
25
2
26
29
26
27
24
3
28
31
30
–
31
4
26
28
27
26
29
Ans: Step 1: Understanding the Scenario
A company has four projects and receives bids from five contractors (A, B, C, D, E). Each
contractor can only be assigned one project, and each project must go to one contractor.
The bids are in thousands of rupees.
Here’s the table again:
Project
A
B
C
D
E
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1
18
25
22
26
25
2
26
29
26
27
24
3
28
31
30
–
31
4
26
28
27
26
29
Notice: Contractor D did not bid for Project 3 (marked “–”). That means D cannot be
assigned Project 3.
Step 2: What Is the Goal?
The company wants to minimize cost (since bids are expenses). So the problem is:
• Assign each project to one contractor.
• Ensure no contractor gets more than one project.
• Minimize the total bid amount.
This is exactly what the Hungarian Method (or assignment method) is designed to solve.
Step 3: Why Is This Complex?
It’s complex because:
• We have more contractors (5) than projects (4).
• One contractor (D) refuses one project.
• We must balance assignments so that every project is covered, but no contractor is
overloaded.
So, we need to standardize the table before applying the method.
Step 4: Standardizing the Table
In assignment problems, the matrix should be square (same number of rows and columns).
Here we have 4 projects × 5 contractors. To make it square, we add a dummy project
(Project 5) with zero cost bids. This dummy project represents “no assignment” for one
contractor, since we have more contractors than projects.
Updated table:
Project
A
B
C
D
E
1
18
25
22
26
25
2
26
29
26
27
24
3
28
31
30
∞
31
4
26
28
27
26
29
5 (dummy)
0
0
0
0
0
Here, “∞” (infinity) is used for D’s missing bid on Project 3, meaning it’s impossible to assign
D to Project 3.
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Step 5: Applying the Assignment Method
The Hungarian Method involves:
1. Row reduction – subtract the smallest value in each row from all elements in that
row.
2. Column reduction – subtract the smallest value in each column from all elements in
that column.
3. Covering zeros – use minimum lines to cover all zeros.
4. Optimal assignment – assign projects to contractors where zeros appear, ensuring
one-to-one assignment.
This process ensures the lowest total cost.
Step 6: Interpreting the Solution
Once solved, the company will know:
• Which contractor should take which project.
• Which contractor should remain unassigned (via dummy project).
• The minimum total cost of assigning all projects.
Diagram: Assignment Problem Setup
Projects → Contractors
-----------------------------------
Each project → exactly one contractor
Each contractor → at most one project
Goal → Minimize total bid cost
Why This Problem Matters
This isn’t just about contractors and projects. Assignment problems appear everywhere:
• Assigning teachers to classes.
• Assigning machines to jobs.
• Assigning workers to shifts.
• Even assigning flights to runways at airports.
The logic is the same: balance tasks and agents to minimize cost or maximize efficiency.
Final Thought
So, the company’s bidding table is really a puzzle of optimal assignment. By converting it
into a square matrix (adding a dummy project and handling missing bids with infinity), we
can apply the Hungarian Method to find the best assignment. The beauty of this problem is
that it shows how mathematics can solve real-world resource allocation challenges—
ensuring fairness, efficiency, and cost-effectiveness.
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SECTION—C
5. Explain Game Theory. Discuss the steps of solving Game Theory problem by Dominance
method by giving suitable examples.
Ans: Game Theory
Imagine you and your friend are playing a game where both of you have to make decisions
without knowing what the other will choose. Your success depends not only on your
decision but also on your friend’s choice. This situation is exactly what Game Theory
studies.
What is Game Theory?
Game Theory is a branch of mathematics and economics that helps us understand strategic
decision-making between two or more players. It is used when:
• There are two or more decision-makers (players)
• Each player has different strategies (choices)
• The result (payoff) depends on all players’ decisions
In simple words:
Game Theory is about thinking ahead—“If I do this, what will the other person do?”
Basic Elements of Game Theory
To understand it clearly, let’s break it down:
1. Players – The participants (e.g., two companies, two players)
2. Strategies – Possible actions (e.g., advertise or not advertise)
3. Payoff – Result or outcome (profit, loss, or points)
4. Game Matrix – A table showing all possible outcomes
Example of a Game Matrix
Let’s consider two companies: A and B. Both can choose either Advertise (A) or Not
Advertise (N).
Company B
A N
Company A ----------------
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A | 5 , 5 | 10 , 2 |
N | 2 ,10 | 6 , 6 |
• The first number = payoff of Company A
• The second number = payoff of Company B
What is the Dominance Method?
The Dominance Method is a simple way to solve game theory problems by eliminating poor
strategies.
A strategy is said to be dominated if there is another strategy that is always better than
it.
Steps to Solve Game Theory Using Dominance Method
Let’s understand step-by-step in a very simple way:
Step 1: Construct the Payoff Matrix
First, write all possible outcomes in a table form (like above).
Step 2: Look for Dominated Rows (for Player A)
• Compare rows (strategies of Player A)
• If one row gives less payoff in all cases, eliminate it
Step 3: Look for Dominated Columns (for Player B)
• Compare columns (strategies of Player B)
• If one column gives worse results, eliminate it
Step 4: Reduce the Matrix
• Continue removing dominated strategies
• Keep simplifying until you get a smaller matrix (usually 2×2 or 1×1)
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Step 5: Find Optimal Strategy
• If one value remains → that is the solution
• If not → further methods may be used (like mixed strategy)
Solved Example Using Dominance Method
Let’s take a simple example:
Player B
B1 B2 B3
A1 3 2 4
A2 2 1 3
A3 5 4 6
Step 1: Check Rows (Player A)
Compare rows:
• A1 = (3, 2, 4)
• A2 = (2, 1, 3)
A2 is smaller than A1 in all cases → A2 is dominated
Remove A2
Step 2: New Matrix
B1 B2 B3
A1 3 2 4
A3 5 4 6
Step 3: Check Rows Again
Compare A1 and A3:
A3 is greater than A1 in all cases → A1 is dominated
Remove A1
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Step 4: Final Matrix
B1 B2 B3
A3 5 4 6
Conclusion
• Only one strategy remains → A3
• Best choice for Player A is A3
Simple Diagram of Dominance Concept
Before Elimination:
A1 → (3,2,4)
A2 → (2,1,3) Dominated
A3 → (5,4,6)
After Elimination:
A3 → Best Strategy
Real-Life Applications of Game Theory
Game Theory is not just theory—it is used everywhere:
• Business competition (pricing, advertising)
• Sports strategies
• 🛡 Military planning
• 🗳 Politics and voting systems
• Negotiations and decision-making
Final Summary
Game Theory helps us understand how people make decisions when outcomes depend on
others. The Dominance Method is a simple and effective way to solve these problems by
removing inferior strategies step-by-step.
Key takeaway:
Always eliminate the weaker options to find the best strategy.
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6. A post office has two clerks, either of whom averages 1.5 minutes per customer
transaction (the service time being distributed exponentially). The arrival rate of the
customers to the post office is one customer per minute. Compute:
(a) The probability that both the clerks would be idle.
(b) The probability that there shall be one customer in the post office.
(c) The probability that there shall be five customers in the post office.
(d) The average number of customers waiting in the queue.
(e) The average number of customers being served.
(f) The average time a customer spends waiting for service.
(g) The average time a customer spends in a post office.
Ans: Step 1: Identify the Parameters
• Arrival rate (λ): 1 customer per minute.
• Service rate (μ): Each clerk serves at a rate of 1 customer per 1.5 minutes =
customers per minute.
• Since there are two clerks, the combined service rate is
customers per
minute.
So:
Step 2: Traffic Intensity
Traffic intensity (ρ) tells us how busy the system is:
So the system is 75% utilized. That means the clerks are busy most of the time, but not
overloaded.
Step 3: Probability of Zero Customers (Both Clerks Idle)
We use the M/M/s formula for
, the probability that no customers are in the system:
Here:
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•
•
So:
So, the probability both clerks are idle is about 14.3%.
Step 4: Probability of One Customer in the Post Office
For , the probability is:
For :
So, about 21.4% chance there is exactly one customer.
Step 5: Probability of Five Customers in the Post Office
For :
For :
So, about 6.8% chance there are five customers.
Step 6: Average Number of Customers Waiting in Queue
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The formula for average queue length
:
So, on average about 1.93 customers wait in line.
Step 7: Average Number of Customers Being Served
That’s simply:
server busy
But easier: average number in service = if servers are busy. Since utilization is 0.75, and
there are 2 clerks:
So, on average 1.5 customers are being served.
Step 8: Average Waiting Time in Queue
Waiting time
:
minutes
So, customers wait about 1.93 minutes before service.
Step 9: Average Time in the Post Office
Total time = waiting time + service time:
minutes
So, each customer spends about 3.43 minutes in the post office.
Diagram: Flow of Customers in M/M/2 System
Arrivals (λ = 1/min) → Queue → Two Clerks (μ = 2/3 each) → Departures
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Final Summary
• (a) Probability both clerks idle = 14.3%
• (b) Probability one customer present = 21.4%
• (c) Probability five customers present = 6.8%
• (d) Average queue length = 1.93 customers
• (e) Average being served = 1.5 customers
• (f) Average waiting time = 1.93 minutes
• (g) Average total time = 3.43 minutes
SECTION—D
7. What do you understand by CPM? State the five steps of the working methodology of
critical path analysis. Can a critical path change during the course of a period?
Ans: What is CPM (Critical Path Method)?
Imagine you are planning to organize a college seminar. There are many tasks to do—
booking a hall, inviting guests, preparing presentations, arranging chairs, etc. Some tasks
can happen at the same time, but some must be done in order.
Now the big question is:
What is the minimum time required to complete everything?
Which tasks are most important and cannot be delayed at all?
This is where CPM (Critical Path Method) comes in.
Simple Meaning:
CPM is a project management technique used to plan, schedule, and control complex
projects by identifying the longest sequence of activities (called the critical path).
The critical path is the longest path of activities in a project.
It determines the minimum time needed to complete the project.
If any task on this path gets delayed, the whole project gets delayed.
Key Idea of CPM
• Every project has many activities.
• Some activities depend on others.
• CPM helps find:
o Which activities are critical
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o Which can be delayed (without affecting the project)
o Total project duration
Diagram to Understand CPM
Here is a simple visual representation of a CPM network:
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In such diagrams:
• Circles/boxes = Activities
• Arrows = Sequence
• Longest path = Critical Path
Five Steps of Critical Path Analysis (CPM Methodology)
Let’s understand the working process step-by-step in a very simple way:
Step 1: Identify All Activities
First, list all the tasks required to complete the project.
Example (Seminar Planning):
• Book hall
• Invite speakers
• Prepare content
• Arrange equipment
This step is like making a to-do list.
Step 2: Determine the Sequence of Activities
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Now decide:
Which task comes first?
Which task depends on another?
Example:
• You can’t start the seminar before booking the hall.
• Invitations must be sent before the event date.
This creates a logical flow of work.
Step 3: Draw the Network Diagram
Now represent all activities in a diagram using nodes and arrows.
This diagram shows:
• Flow of tasks
• Dependencies
• Parallel activities
Think of it like a road map of your project.
Step 4: Estimate Time for Each Activity
Assign time to each task.
Example:
• Booking hall → 2 days
• Sending invitations → 3 days
• Preparing content → 5 days
This helps calculate total project duration.
Step 5: Identify the Critical Path
Now calculate:
• All possible paths
• Total time of each path
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The longest path = Critical Path
Important:
• Activities on this path have zero flexibility (no delay allowed).
• Other activities may have some slack time.
What is Slack Time?
Slack (or float) means:
Extra time available for an activity without delaying the project.
Example:
If a task can be delayed by 2 days without affecting the project, its slack = 2 days.
Can the Critical Path Change?
Yes, the critical path can change during the project.
Let’s understand why:
Reasons for Change in Critical Path
1. Delay in Activities
If a non-critical activity gets delayed beyond its slack time, it can become critical.
2. Faster Completion
If a critical activity finishes earlier, another path may become longer and become critical.
3. Changes in Project Plan
Adding or removing tasks can change the network structure.
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4. Resource Issues
Lack of manpower or materials may delay certain activities.
Simple Example:
Imagine two paths:
• Path A → 10 days (Critical Path)
• Path B → 8 days
If Path B gets delayed by 3 days → total becomes 11 days
Now Path B becomes the new critical path
Final Understanding
Let’s summarize everything:
CPM is used to:
• Plan projects
• Schedule activities
• Control delays
Critical Path:
• Longest path in the project
• Determines total project time
• No delay allowed
Five Steps:
1. Identify activities
2. Determine sequence
3. Draw network diagram
4. Estimate time
5. Find critical path
Important Point:
Yes, the critical path can change during the project depending on delays, changes, or
execution speed.
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Real-Life Connection
Think of CPM like planning your exam preparation:
• Subjects = Activities
• Study order = Sequence
• Time per subject = Duration
• Final exam date = Deadline
The most time-consuming subjects become your critical path!
8. A project consists of eight activities with the following time estimates:
Activity
Immediate Predecessor
Optimistic
Most Likely
Pessimistic
A
–
1
1
7
B
–
1
4
7
C
–
2
2
8
D
A
1
1
1
E
B
2
5
14
F
C
2
5
8
G
D & E
3
6
15
H
F & G
1
2
3
Required:
(a) Draw PERT network.
(b) Find the expected time for each activity.
(c) Determine the earliest event times and latest allowable times.
(d) Determine the critical path.
(e) Determine the total slack for each activity.
(f) What is the probability that the project will be completed in
(i) 22 days (ii) 18 days (iii) 19 days?
(g) What project duration will have 95% chance of completion?
(h) If the average duration for activity F increases to 14 days,
what will be its effect on the expected project completion time which will have 95%
confidence?
Ans: Step 1: The Data Table
Here’s the project data:
Activity
Immediate Predecessor
Optimistic
Most Likely
Pessimistic
A
–
1
1
7
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B
–
1
4
7
C
–
2
2
8
D
A
1
1
1
E
B
2
5
14
F
C
2
5
8
G
D & E
3
6
15
H
F & G
1
2
3
Step 2: Expected Time for Each Activity
PERT uses the formula:
where O = optimistic, M = most likely, P = pessimistic.
• A:
• B:
• C:
• D:
• E:
• F:
• G:
• H:
So expected times: A=2, B=4, C=3, D=1, E=6, F=5, G=7, H=2.
Step 3: Draw the PERT Network
The network flows like this:
• Start → A → D → G → H → End
• Start → B → E → G → H → End
• Start → C → F → H → End
This shows three parallel paths merging into G and H.
Step 4: Earliest Event Times (Forward Pass)
• Start = 0
• A finishes at 2 → D finishes at 3 → G finishes at 10 → H finishes at 12.
• B finishes at 4 → E finishes at 10 → G (already 10).
• C finishes at 3 → F finishes at 8 → H (needs both F=8 and G=10, so H starts at 10) →
H finishes at 12.
So project completion time = 12 days (expected).
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Step 5: Latest Event Times (Backward Pass)
Work backward from End (12):
• H must finish by 12 → start at 10.
• G must finish by 10.
• F must finish by 10 (since H starts at 10).
• E must finish by 10.
• D must finish by 3.
• A must finish by 2.
• B must finish by 4.
• C must finish by 3.
Step 6: Critical Path
Critical path = longest path through the network. Let’s check:
• Path A–D–G–H = 2 + 1 + 7 + 2 = 12
• Path B–E–G–H = 4 + 6 + 7 + 2 = 19
• Path C–F–H = 3 + 5 + 2 = 10
So the critical path is B–E–G–H = 19 days. That means the expected project duration is 19
days, not 12 (because the longest path determines completion).
Step 7: Slack for Each Activity
Slack = Latest Start – Earliest Start.
• Activities on the critical path (B, E, G, H) have zero slack.
• Others (A, C, D, F) have positive slack (they can be delayed without affecting
completion).
Step 8: Probability Calculations
PERT uses variance:
Compute variance for critical path activities:
• B: , variance = 1
• E: , variance = 4
• G: , variance = 4
• H: , variance ≈ 0.11
Total variance = 9.11, σ ≈ 3.02. Expected duration = 19 days.
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Now probabilities:
• For 22 days: . Probability ≈ 84%.
• For 18 days: . Probability ≈ 37%.
• For 19 days: . Probability ≈ 50%.
Step 9: Duration for 95% Confidence
We want Z ≈ 1.645 (for 95%).
days
So, 24 days gives 95% chance of completion.
Step 10: Effect of Increasing Activity F to 14 Days
If F’s expected time rises to 14, the path C–F–H becomes: 3 + 14 + 2 = 19. Now there are two
critical paths: B–E–G–H and C–F–H, both 19 days. Variance increases (since F’s variance is
larger), so the 95% confidence duration will be longer than 24 days. The project becomes
riskier.
Diagram: PERT Network (simplified)
Start
├── A (2) → D (1) → G (7) → H (2) → End
├── B (4) → E (6) ─┘
└── C (3) → F (5) ─┘
Final Summary
• Expected times: A=2, B=4, C=3, D=1, E=6, F=5, G=7, H=2.
• Critical path: B–E–G–H = 19 days.
• Slack: Zero for critical path, positive for others.
• Probabilities: 22 days ≈ 84%, 18 days ≈ 37%, 19 days ≈ 50%.
• 95% confidence duration ≈ 24 days.
• If F increases to 14 days, project risk rises and completion time at 95% confidence
becomes longer.
“This paper has been carefully prepared for educaonal purposes. If you noce any
mistakes or have suggesons, feel free to share your feedback.”
