GNDU Answer PAPERS 2025
BBA 4
th
SEMESTER
Paper–BBA04008T : OPERATIONS RESEARCH
Time Allowed: 3 Hours Maximum Marks:100
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. What do you mean by Operaons Research ? Discuss its scope in detail.
Ans. Operaon Research is a quantave approach to decision-making. It employs
mathemacal models and analycal techniques to opmise resource allocaon, improve
eciency, and enhance overall performance. It involves the following:
Problem Denion: Clearly idenfying the problem to be solved.
Model Formulaon: developing a mathemacal model that represents the problem.
Model Soluon: Employing algorithms to solve the model and obtain opmal soluons.
Implementaon: translang the soluons into praccal acons.
OR helps organisaons make informed decisions by providing a structured and analycal
framework for evaluang dierent opons and choosing the best course of acon.
Importance of Operaon Research
The given below are the benets to organizaons oered by Operaon Research:
Improved Decision-Making: OR provides quantave insights that support informed
decision-making.
Increased Eciency: By opmising resource allocaon and processes, OR can lead to
signicant eciency gains.
Cost Reducon: OR can help idenfy cost-saving opportunies and reduce waste.
Enhanced Performance: OR can improve the overall performance of organisaons by
opmising key metrics.
Risk Management: OR can be used to assess risks and develop strategies for migang them.
Scope of Operaon Research
In recent years of organized development, OR has entered successfully in many dierent
areas of research. It is useful in the following various important elds
In agriculture
With the sudden increase of populaon and resulng shortage of food, every country is
facing the problem of
Opmum allocaon of land to a variety of crops as per the climac condions
Opmum distribuon of water from numerous resources like canal for irrigaon purposes
Hence there is a requirement of determining best policies under the given restricons.
Therefore a good quanty of work can be done in this direcon.
In nance
In these recent mes of economic crisis, it has become very essenal for every government
to do a careful planning for the economic progress of the country. OR techniques can be
producvely applied
To determine the prot plan for the company
To maximize the per capita income with least amount of resources
To decide on the best replacement policies, etc
In industry
If the industry manager makes his policies simply on the basis of his past experience and a
day approaches when he gets rerement, then a serious loss is encounter ahead of the
industry. This heavy loss can be right away compensated through appoinng a young
specialist of OR techniques in business management. Thus OR is helpful for the industry
director in deciding opmum distribuon of several limited resources like men, machines,
material, etc to reach at the opmum decision.
In markeng
With the assistance of OR techniques a markeng administrator can decide upon
Where to allocate the products for sale so that the total cost of transportaon is set to be
minimum
The minimum per unit sale price
The size of the stock to come across with the future demand
How to choose the best adversing media with respect to cost, me etc?
How, when and what to buy at the minimum likely cost?
In personnel management
A personnel manager can ulize OR techniques
To appoint the highly suitable person on minimum salary
To know the best age of rerement for the employees
To nd out the number of persons appointed in full me basis when the workload is
seasonal
In producon management
A producon manager can ulize OR techniques
To calculate the number and size of the items to be produced
In scheduling and sequencing the producon machines
In compung the opmum product mix
To choose, locate and design the sites for the producon plans
In L.I.C
OR approach is also applicable to facilitate the L.I.C oces to decide
What should be the premium rates for a range of policies?
How well the prots could be allocated in the cases of with prot policies?
Role of Operaons Research in Decision-Making
The Operaon Research may be considered as a tool which is employed to raise the
eciency of management decisions. OR is the objecve complement to the subjecve
feeling of the administrator (decision maker). Scienc method of OR is used to comprehend
and explain the phenomena of operang system.
2. 2. (a) Solve the following LPP problem by Simplex Method :
Minimize Z = 4x₁ + x₂
Subject to :
3x₁ + 4x₂ ≥ 20
−x₁ − 5x₂ ≤ −15
where x₁, x₂ ≥ 0.
Ans:
(b) Give steps to solve LPP by Two Phase Simplex Method with help of illustraon.
Ans. The Two-Phase Simplex Method When a basic feasible soluon is not readily available,
the two-phase simplex method may be used as an alternave to the Big M method. In the
two-phase simplex method, we add arcial variables to the same constraints as we did in
the Big M method. Then we nd a bfs to the original LP by solving the Phase I LP. In the
Phase I LP, the objecve funcon is to minimize the sum of all arcial variables. At the
compleon of Phase I, we reintroduce the original LP’s objecve funcon and determine the
opmal soluon to the original LP. The following steps describe the two-phase simplex
method. Note that steps 1–3 for the two-phase simplex are idencal to steps 1–4 for the Big
M method.
Steps
1) Modify the constraints so that the right-hand side of each constraint is nonnegave. This
requires that each constraint with a negave right-hand side be mulplied through by _1.
2) Idenfy each constraint that is now (aer step 1) ≥ or = constraint. In step 3, we will add
an arcial variable to each constraint.
3) Convert each inequality constraint to the standard form. If constraint i is ≤ constraint, then
add a slack variable si. If constraint i is ≥ constraint, subtract an excess variable ei.
4) If (aer step 2) constraint i is ≥ or = constraint, add an arcial variable ai.
5) For now, ignore the original LP’s objecve funcon. Instead solve an LP whose objecve
funcon is min W = (sum of all the arcial variables). This is called the Phase I LP.
The act of solving the Phase I LP will force the arcial variables to be zero. Because each ai
≥ 0,
solving the Phase I LP will result in one of the following three cases:
Case 1
The opmal value of W is greater than zero. In this case, the original LP has no feasible
soluon.
Case 2
The opmal value of W is equal to zero, and no arcial variables are in the opmal Phase I
basis. In this case, we drop all columns in the opmal Phase I tableau that corresponds to
the arcial variables. We BY: Dr. Hiba G. Fareed 51 now combine the original objecve
funcon with the constraints from the opmal Phase I tableau. This yields the Phase II LP.
The opmal soluon to the Phase II LP is the opmal soluon to the original LP.
Case 3
The opmal value of W is equal to zero and at least one arcial variable is in the opmal
Phase I basis. In this case, we can nd the opmal soluon to the original LP if at the end of
Phase I we drop from the opmal Phase I tableau all nonbasic arcial variables and any
variable from the original problem that has a negave coecient in objecve row of the
opmal Phase I tableau
SECTION–B
3. Dene Dual Problem. Discuss general rules of converng primal into its dual.
Ans.
A dual problem is a complementary linear programming (LP) model derived from an original
problem, sharing the same data but reversing the roles of constraints and variables. It oers
an alternave, oen more ecient, method for solving LPPs by turning maximizaon into
minimizaon (or vice versa), where dual opmal values align with primal soluons.
To convert a primal linear programming problem (LPP) to its equivalent dual problem,
follow these steps:
Idenfy the primal problem: Write down the primal problem in standard form. The primal
problem should be a maximizaon or minimizaon problem with constraints.
Formulate the dual problem: The dual problem is derived from the primal problem by
interchanging the roles of the constraints and the objecve funcon. The coecients of the
constraints in the primal problem become the coecients of the objecve funcon in the
dual problem, and vice versa.
Handle equality constraints: When there is an equality constraint in the primal problem, the
corresponding variable in the dual problem will be unrestricted in sign (i.e., it can take any
real value).
Write the dual problem: Use the following rules to write the dual problem:
If the primal problem is a maximizaon problem, the dual will be a minimizaon problem,
and vice versa.
The number of variables in the dual problem will be equal to the number of constraints in
the primal problem, and the number of constraints in the dual problem will be equal to the
number of variables in the primal problem.
The coecients of the objecve funcon in the primal problem become the right-hand side
constants in the dual problem, and the right-hand side constants in the primal problem
become the coecients of the objecve funcon in the dual problem.
Example:
Consider the following primal problem:
Maximize Z=3x1+2x2
subject to:
x1+x2=4
2x1+x2≤6
x1,x2≥0
Steps to convert to dual problem:
Idenfy the primal problem: The given primal problem is a maximizaon problem with two
constraints and two variables.
Formulate the dual problem: The dual problem will be a minimizaon problem with two
variables and two constraints.
Handle equality constraint: The rst constraint x1+x2=4 is an equality constraint, so the
corresponding dual variable y1 will be unrestricted in sign.
Write the dual problem:
Write the dual problem:
Minimize W=4y1+6y2
subject to:
y1+2y2≥3
y1+y2≥2
y2≥0
y1 is unrestricted in sign.
Therefore, the dual problem is:
Minimize W=4y1+6y2
subject to:
y1+2y2≥3
y1+y2≥2
y2≥0
y1 is unrestricted in sign.
4. (a) Solve the following assignment problems having the following cost element :
1
2
3
4
A
85
50
30
40
B
90
40
70
45
C
70
60
60
50
D
75
45
35
55
Soluon
b. (b) Explain the following terms :
(1) Inial Feasible Soluon
(2) Unbalanced Transportaon Problem.
Soluon
1. Inial basic feasible soluons (IBFS) in transportaon problems (Operaon Research)
are foundaonal topics in B.Com studies, focusing on nding a starng, non-negave
allocaon that sases demand and supply constraints.
Methods for Inial Feasible Soluon
North West Corner Rule (NWCR): Starts at the top-le cell, oering the simplest method but
oen yielding a higher cost.
Least Cost Method (LCM): Focuses on allocang to cells with the lowest transportaon costs
rst, resulng in a beer inial soluon than NWCR.
Vogel's Approximaon Method (VAM): Evaluates penalty costs to make beer-informed
allocaons. It usually provides the closest approximaon to the opmal soluon.
Unbalanced Transportaon Problem
Unbalanced transportaon problem is dened as a situaon in which supply and demand
are not equal. A dummy row or a dummy column is added to this type of problem,
depending on the necessity, to make it a balanced problem. The problem can then be
addressed in the same way as the balanced problem.
An unbalanced transportaon problem is converted into a balanced transportaon problem
by introducing a dummy origin or a dummy desnaons which will provide for the excess
availability or the requirement the cost of transporng a unit from this dummy origin (or
dummy desnaon) to any place is taken to be zero.
SECTION–C
5. Discuss meaning and types of inventory. Also, discuss Economic lot size models with
example.
Soluon Meaning of Inventory
Inventory refers to the stock of goods, materials, or resources that a business keeps on hand
to meet future demand. It includes raw materials, work-in-progress, nished goods, and
other items required for producon and sales.
Inventory is essenal for ensuring smooth producon, avoiding shortages, and meeng
customer demand on me. However, excessive inventory increases holding costs, so proper
management is necessary.
Types of Inventory
1. Raw Materials
Raw materials are the basic inputs used in the producon process. These are purchased
from suppliers and transformed into nished goods. For example, steel in automobile
manufacturing or coon in texle producon.
2. Work-in-Progress (WIP)
Work-in-progress inventory includes parally completed goods that are sll undergoing
producon. These items are not yet ready for sale and are in dierent stages of compleon.
3. Finished Goods
Finished goods are the nal products that are ready for sale to customers. These are kept in
warehouses unl they are sold or distributed.
4. Stores and Spares
These include maintenance items, spare parts, lubricants, and other supplies required for
smooth funconing of machinery and operaons.
5. Pipeline Inventory
This refers to goods that are in transit from one locaon to another, such as from supplier to
factory or from factory to warehouse.
Economic Lot Size (EOQ) Model
The Economic Order Quanty (EOQ) model determines the opmal quanty of items a rm
should order each me so that the total cost of ordering and holding inventory is minimized.
Assumpons of EOQ Model
1. Demand is known and constant
2.Lead me is xed
3. Ordering cost and holding cost are constant
4. No stockouts are allowed
5. Purchase price per unit is constant
Example of EOQ
Suppose:
Annual demand (D) = 1000 units
Ordering cost (S) = ₹50 per order
Holding cost (H) = ₹2 per unit per year
Applying the formula:
EOQ=
√
2∗1000∗50
2
= 223.6 unit or 224 units
Interpretaon
The rm should order approximately 224 units each me to minimize total inventory cost.
Conclusion
Inventory management is crucial for maintaining a balance between demand and supply.
The EOQ model helps rms determine the most economical order quanty, thereby
reducing total costs and improving eciency.
6. Reduce the following game by Dominance property and solve it :
B1
B2
B3
B4
B5
A1
2
4
3
8
5
A2
4
5
2
6
7
A3
7
6
8
7
6
A4
3
1
7
4
2
Soluon
SECTION–D
7. The following table gives the data for the acvies of a small project :
Job
Opmisc
Most
likely
Pessimisc
1–2
1
4
7
1–3
5
10
15
2–4
3
3
3
2–6
1
4
7
3–4
10
15
26
3–5
2
4
6
4–5
5
5
5
5–6
2
5
8
(i) Draw the network and nd the expected project compleon me.
(ii) Find crical paths.
(iii) Earliest expected and latest expected me for each event.
(iv) Determine scheduling of acvies and compute various oats.
Soluon:
8. Dene the following terms :
(a) Event and Acvity.
(b) Errors in Network Logic.
(c) Fulkerson’s rule to Numbering of Events.
(d) How crical path is computed in Networking problems ?
Soluon a)
b) Errors in Network Logic
Errors in network logic refer to mistakes made while construcng a project network diagram.
These errors can lead to incorrect scheduling and analysis.
The main types of errors are:
Looping Error:
This occurs when a sequence of acvies forms a closed loop. It creates confusion as no
acvity can logically follow itself.
Dangling Error:
This happens when an acvity is not properly connected to the network, meaning it is le
hanging without a proper start or end event.
Redundancy Error:
Occurs when unnecessary acvies or relaonships are included, making the network more
complex than required without adding value.
c) Fulkerson’s Rule for Numbering of Events
Fulkerson’s Rule is used to assign numbers to events in a network diagram in a logical order.
Steps involved:
Idenfy the starng event and assign it number 1.
Move from le to right and assign numbers to events such that:
An event can be numbered only when all its preceding events have been numbered.
Ensure that:
No two events have the same number.
The numbering follows a logical sequence from start to end.
This rule helps in avoiding confusion and makes calculaons (like earliest and latest mes)
easier.
d. The crical path is the longest path in a project network and determines the minimum
me required to complete the project.
Steps to compute the crical path:
1. Construct the Network Diagram
2. Draw the network showing all acvies and their dependencies.
Forward Pass (Earliest Time Calculaon)
Start from the rst event.
Calculate Earliest Start Time (EST) and Earliest Finish Time (EFT).
3. Backward Pass (Latest Time Calculaon)
Start from the last event and move backward.
Calculate Latest Start Time (LST) and Latest Finish Time (LFT).
4. Calculate Slack (Float)
Slack = LST − EST (or LFT − EFT)
Acvies with zero slack are crical.
5. Idenfy Crical Path
The path connecng all crical acvies (with zero slack) is the crical path.
It represents the longest duraon path in the network.
