This page contains detailed solutions of MBA (QTM) – Operations Research Previous Year Question Paper 2022. All answers are written in simple and exam-oriented format suitable for 2 to 10 marks university questions.
University: Dr. A.P.J. Abdul Kalam Technical University (AKTU)
Course: MBA – Quantitative Techniques for Management (QTM)
Year: 2022
This paper covers Linear Programming, Transportation Problem, Assignment Model, Game Theory, Queuing Theory, Replacement Model, CPM and Decision Tree Analysis.
Operations Research (OR) is a scientific approach to decision-making using mathematical models and analytical techniques.
Conclusion: OR helps in optimal use of limited resources in managerial decisions.
Uncertainty refers to a situation where future outcomes are unknown and probabilities cannot be assigned.
Example: Launching a new product without demand history.
Uncertainty increases risk and makes decision-making complex.
Linear Programming Problem (LPP) is a mathematical technique to maximize or minimize a linear objective function subject to linear constraints.
Example: Maximizing profit under labor and material constraints.
LPP ensures optimal allocation of resources.
Vogel’s Approximation Method (VAM) is used to obtain an initial feasible solution for a transportation problem.
VAM generally gives a better initial solution than NW Corner method.
The Assignment Model is a special case of transportation problem where one-to-one allocation is done.
Objective: Minimize cost or maximize profit.
Ensures optimal job allocation.
A competitive situation where gain of one player equals loss of the other.
Used in competitive business strategies.
Sequencing problem determines the optimal order of jobs on machines to minimize total time.
Improves production efficiency.
A Queue Model studies waiting lines to minimize waiting time and service cost.
Example: Bank counter waiting system.
Balances service efficiency and cost.
Replacement concept determines optimal time to replace equipment to minimize cost.
Helps reduce maintenance and breakdown cost.
Critical Path Method (CPM) is a network technique used in project planning and control.
Ensures timely project completion.
Operations Research (OR) provides scientific and quantitative techniques to solve complex industrial management problems. Two important OR methods widely used in industries are Linear Programming and Transportation Model.
Linear Programming is a mathematical optimization technique used to maximize or minimize a linear objective function subject to linear constraints.
Basic Components of LP:
Industrial Applications:
Example: Agar factory ke paas limited labour hours aur raw material ho, LP determine karta hai kaunsa product kitna produce kare taki maximum profit mile.
Advantages:
The Transportation Model is used to determine the least cost method of transporting goods from several sources to several destinations.
Objective: Minimize total transportation cost.
Methods Used:
Industrial Applications:
Example: Multiple factories se goods ko multiple warehouses tak bhejna at minimum transportation cost.
Conclusion: Linear Programming and Transportation models help industries reduce cost, increase efficiency and improve managerial decision-making.
The North-West Corner Method (NWCM) is a technique used to obtain an Initial Basic Feasible Solution (IBFS) of a transportation problem.
Step 1: Start from North-West Corner
Transportation table ke top-left cell se allocation start karo.
Step 2: Allocate Minimum of Supply and Demand
Allocation = min(Supply, Demand)
(Hinglish: jo value chhoti ho supply ya demand usko allocate kar do)
Step 3: Adjust Supply and Demand
Allocated quantity ko supply aur demand se subtract karo.
Step 4: Move to Next Cell
If supply becomes zero → move down.
If demand becomes zero → move right.
Step 5: Repeat the Process
Continue allocation until all supply and demand are satisfied.
Important Condition:
Total allocations must be = m + n − 1 (non-degenerate solution).
Limitation: This method ignores transportation cost, so it may not give optimal solution.
The Hungarian Algorithm is used to solve assignment problems where the objective is to minimize cost or maximize profit.
Step 1: Row Reduction
Har row ka smallest element find karo aur usko puri row se subtract karo.
Step 2: Column Reduction
Har column ka smallest element subtract karo.
Step 3: Cover All Zeros
Minimum number of horizontal and vertical lines draw karo to cover all zeros.
Step 4: Optimality Test
If number of lines = order of matrix → optimal solution found.
If not, smallest uncovered element subtract karke matrix adjust karo.
Step 5: Make Assignment
Independent zeros choose karo (ek row aur ek column me sirf ek assignment).
Advantages:
Conclusion: Hungarian method ensures minimum total cost assignment efficiently.
Johnson’s Algorithm is used for sequencing n jobs on two machines to minimize total elapsed time.
Benefits:
Project Management involves planning, scheduling and controlling activities to complete a project within specified time and cost limits.
A network diagram represents project activities and their relationships graphically.
Conclusion: CPM and PERT are powerful tools in project management for effective planning and control.
Operations Research (OR) provides scientific and quantitative techniques that help managers take rational and optimal decisions. The three important OR techniques used in managerial decision-making are:
Linear Programming is a mathematical optimization technique used to maximize or minimize a linear objective function subject to linear constraints.
Main Elements:
Managerial Applications:
Example: Agar ek company do products banati hai aur labour hours limited hain, LP determine karta hai kitna production kare taki maximum profit mile.
Importance:
Transportation Model: Used to minimize cost of distributing goods from multiple sources to multiple destinations.
Applications:
Assignment Model: Special case of transportation problem where one-to-one assignment is made.
Applications:
Importance: Reduces operational cost and improves efficiency.
This technique is used when decisions are taken under conditions of certainty, risk, and uncertainty.
Applications:
Importance:
Conclusion: Linear Programming, Transportation/Assignment Models and Decision Tree techniques are powerful tools that enable managers to take scientific, cost-effective and optimal decisions.
A Decision Tree is a graphical representation of decision alternatives and possible outcomes including probabilities and payoffs. It is mainly used in decision-making under risk and uncertainty.
Step 1: Define the problem clearly.
Step 2: Identify possible alternatives (e.g., launch product or not).
Step 3: Assign probabilities to different outcomes.
Step 4: Calculate Expected Monetary Value (EMV).
Formula:
EMV = Σ (Probability × Payoff)
A company is deciding whether to launch a new product.
EMV Calculation:
EMV = (0.6 × 5,00,000) + (0.4 × -2,00,000)
= 3,00,000 − 80,000
= ₹2,20,000
Since EMV is positive, launching the product is a better decision.
Conclusion: Decision Tree is a powerful managerial tool that helps in selecting the best alternative by evaluating probabilities and expected values systematically.
Maximize
Z = 6L₁ + 11L₂
Subject to:
1) 2L₁ + L₂ ≤ 104
2) L₁ + 2L₂ ≤ 76
3) L₁ ≥ 0 , L₂ ≥ 0
This Linear Programming Problem is solved using the Graphical Method because it contains only two decision variables.
2L₁ + L₂ = 104
L₁ + 2L₂ = 76
(Boundary lines draw karne ke liye inequality ko equal banate hain.)
For 2L₁ + L₂ = 104
Points: (0,104) and (52,0)
For L₁ + 2L₂ = 76
Points: (0,38) and (76,0)
2L₁ + L₂ = 104 ...(1)
L₁ + 2L₂ = 76 ...(2)
2L₁ + 4L₂ = 152
Subtract equation (1):3L₂ = 48
L₂ = 16
Substitute in (1):2L₁ + 16 = 104
L₁ = 44
Intersection Point = (44,16)
Z(0,0) = 0
Z(52,0) = 312
Z(0,38) = 418
Z(44,16) = 440
Maximum Z = 440
L₁ = 44 , L₂ = 16
The optimal solution lies at the intersection point of the two constraints.
The Graphical Method is used to solve Linear Programming Problems (LPP) when there are only two decision variables. It provides a visual representation of constraints and helps identify the optimal solution from the feasible region.
Define:
(Sabse pehle problem ko mathematical form me convert karte hain.)
Replace ≤ or ≥ signs with equal (=) sign to draw boundary lines.
The feasible region is the common shaded area that satisfies all constraints.
(Ye area first quadrant me hota hai kyunki variables non-negative hote hain.)
Determine coordinates of all intersection points of constraint lines.
Substitute each corner point into objective function.
The point giving maximum or minimum value is the optimal solution.
The Graphical Method is a fundamental technique in Linear Programming that helps managers determine the optimal solution visually by analyzing feasible region and corner points.
m1 → 12 8 7 8
m2 → 6 6 4 8
m3 → 3 5 7 4
m4 → 1 3 5 4
Har row ka smallest element subtract karte hain.
Ab har column ka smallest element subtract karte hain.
Minimum number of horizontal and vertical lines draw kiye to cover all zeros.
Since number of lines = 4 (order of matrix), optimal solution exists.
Independent zeros choose karte hain (ek row aur ek column me ek assignment).
Total Cost = 8 + 4 + 4 + 1 = 17
Optimal Assignment: m1–J2, m2–J3, m3–J4, m4–J1
Minimum Total Cost = 17
Game Theory is a mathematical technique used to study competitive situations where two or more players take strategic decisions.
In a Two-Person Zero-Sum Game, the gain of one player is equal to the loss of the other player.
Total payoff = Zero.
Rows represent Player A strategies and columns represent Player B strategies.
Player A finds minimum payoff in each row and selects maximum among them.
Player B finds maximum payoff in each column and selects minimum among them.
If Maximin = Minimax → Saddle point exists and pure strategy solution is obtained.
If not equal → Mixed strategy method is used.
B1 B2
A1 4 2
A2 3 5
Row minimums → 2 and 3 → Maximin = 3
Column maximums → 4 and 5 → Minimax = 4
Since Maximin ≠ Minimax, no saddle point exists.
Game Theory provides a logical framework to analyze competitive situations and select optimal strategies.
Arrival Rate (λ):
λ = 20 claims per 8 hours
λ = 20 / 8 = 2.5 claims per hour
Service Rate (μ):
Average service time = 40 minutes = 40/60 hours = 2/3 hour
μ = 1 / (2/3) = 1.5 per hour (per adjuster)
Total service rate (for 3 servers):
sμ = 3 × 1.5 = 4.5 per hour
ρ = λ / (sμ)
ρ = 2.5 / 4.5
ρ = 0.556
This means each adjuster is busy 55.6% of the time.
Working hours per week = 5 × 8 = 40 hours
Busy hours = 0.556 × 40 = 22.24 hours
An adjuster spends approximately 22.24 hours per week attending claimants.
First find λ/s:
λ/s = 2.5 / 3 = 0.833
W = 1 / (μ − λ/s)
W = 1 / (1.5 − 0.833)
W = 1 / 0.667
W ≈ 1.5 hours
On average, a claimant spends approximately 1.5 hours in the branch office.
Poisson Distribution is used to describe the probability of a given number of arrivals occurring in a fixed time interval.
Formula:
P(x) = (e^−λ × λ^x) / x!
Where:
If average arrivals λ = 4 per hour, find probability of exactly 2 arrivals.
P(2) = (e^−4 × 4²) / 2!
= (e^−4 × 16) / 2
= 8e^−4
Poisson distribution plays a fundamental role in modeling random arrivals in queuing systems and helps in efficient service system design.
Replacement problems arise when machines, equipment or components become inefficient or fail. In case of sudden failure, items fail completely without prior warning.
Examples include electric bulbs, fuses, electronic parts and machine components.
Compare:
The policy with minimum average cost is selected.
Replacement analysis helps management decide the most economical time and method for replacing assets that fail suddenly.
ES = 0
EF = 0 + 10 = 10
ES = 0
EF = 10
Starts after 5 days of v1
ES = 5
EF = 5 + 6 = 11
Depends on:
Maximum of (4, 8, 6) = 8
ES = 8
EF = 8 + 8 = 16
Maximum = 10
ES = 10
EF = 10 + 12 = 22
Final activities finish at:
Project completion time = 22 days
The longest path is:
v1 → v5
Total Duration = 10 + 12 = 22 days
Critical Path = v1 → v5
Minimum Completion Time = 22 days
Critical path activities have zero slack and determine the total project duration.