MCA 251 Operational Research Notes cover all 4 units of this GGSIPU MCA Semester 3 open elective in a 50-page printed book: 22 solved numericals, 40 short-answer questions with model answers, 35 long practice questions and 4 revision cards. They follow the MCA syllabus for affiliated institutes, Academic Year 2025-26 onwards.
Written by Prateek Talwar, Founder, Unnati Educations · Reviewed by Ritik Rawat, B.Tech, Senior Full Stack Developer · Checked against the official MCA syllabus PDF (approved 01 Aug 2025) on 13 Oct 2026 · Updated 14 October 2026 · Editorial policy
Scheme: MCA syllabus for affiliated institutes, A.S. 2025-26 | Applies to: Academic Year 2025-26 onwards | Verified against: syllmcaaff050825900p.pdf, approved 01 Aug 2025
MCA 251 Operational Research at a glance: GGSIPU MCA Sem 3 credits, units and exam pattern
MCA 251 (printed MCA-251 in the official PDF) is one of four Open Elective II choices, and it sits with the other GGSIPU MCA Semester 3 papers in a 27-credit term.
| Code and title | MCA-251 Operational Research |
|---|---|
| Course and semester | MCA, Semester 3 |
| Category and credits | Open Elective II · 3 credits · L 3, T/P nil |
| Marks | 100: 40 internal + 60 end-term |
| Pass marks | 40 out of 100 (scheme note 9) |
| Applies to | Affiliated institutes, Academic Year 2025-26 onwards |
| End-term pattern | Q1 compulsory, short answers from the whole syllabus, 12 marks; then one of two questions per unit, 12 marks each (12 + 48 = 60 marks) |
| Units and hours | 4 units: 10 + 10 + 10 + 10 = 40 hours (numbered I to IV, no official titles) |
| Paired lab | None (Sem 3 labs MCA-265 and MCA-267 follow the core electives) |
Scheme note 3 matters for electives: if fewer than 20 students pick one, it may run through a MOOC instead of classes. Note 8 places the programme under Ordinance 11, the source for re-appear rules.
MCA 251 Operational Research Notes unit-wise: 22 solved numericals and 40 short answers across 4 units
Each unit in the 50-page book ends with short answers, long questions and a revision card. The unit titles below are the notes' own, and one page per unit is in the sample PDF above. Notes for the other papers sit under GGSIPU MCA notes.
MCA 251 Unit 1: Introduction to OR and Linear Programming
Unit 1 (pages 3 to 16) covers OR's history and model types, then linear programming: formulation, graphical method and its special cases, simplex, Big-M, two-phase, duality with shadow prices and sensitivity ranges. One product-mix problem runs through all of it and is then solved in Excel Solver, LINGO and TORA. It has 6 solved examples, 10 short answers and 10 long questions.
| Basis | Big-M method | Two-phase method |
|---|---|---|
| Stages | Single stage | Phase I + Phase II |
| Penalty | Uses a large penalty M | No M used |
| On a computer | Rounding errors possible | Computationally more stable |
| By hand | Fewer tables | Slightly more work |
Quick look: Big-M vs two-phase method (from page 13)
- Basis
- Stages
- Big-M method
- Single stage
- Two-phase method
- Phase I + Phase II
- Basis
- Penalty
- Big-M method
- Uses a large penalty M
- Two-phase method
- No M used
- Basis
- On a computer
- Big-M method
- Rounding errors possible
- Two-phase method
- Computationally more stable
- Basis
- By hand
- Big-M method
- Fewer tables
- Two-phase method
- Slightly more work
MCA 251 Unit 2: Transportation and Assignment Problems
Unit 2 (pages 17 to 29) solves one 3 × 4 cost table for a starting solution three ways, then takes it to the optimum with MODI. Unbalanced, maximisation, degeneracy (ε) and prohibited routes follow, then the Hungarian method, unbalanced and profit cases, and the travelling salesperson problem. It has 5 solved examples, 10 short answers and 9 long questions.
| Method | Total cost |
|---|---|
| North-West Corner | ₹1015 |
| Least Cost | ₹814 |
| Vogel's Approximation | ₹779 (MODI optimum: ₹743) |
Quick look: starting cost by method on the same problem (from page 20)
- Method
- North-West Corner
- Total cost
- ₹1015
- Method
- Least Cost
- Total cost
- ₹814
- Method
- Vogel's Approximation
- Total cost
- ₹779 (MODI optimum: ₹743)
MCA 251 Unit 3: Queuing Theory and PERT/CPM
Unit 3 (pages 30 to 40) covers arrivals, balking and reneging, queue discipline, Kendall's notation, Little's law, the M/M/1 and M/M/c models and Erlang service times; then project scheduling, where one network is worked through CPM floats, PERT probability and crashing. It has 5 solved examples, 10 short answers, 9 long questions and 2 diagrams.
| Measure | Result |
|---|---|
| P0 (counter idle) | 0.25 |
| Ls, in system | 3 customers |
| Lq, in queue | 2.25 customers |
| Ws, time in system | 30 minutes |
Quick look: M/M/1 counter, λ = 6/hour, μ = 8/hour, ρ = 0.75 (from page 34)
- Measure
- P0 (counter idle)
- Result
- 0.25
- Measure
- Ls, in system
- Result
- 3 customers
- Measure
- Lq, in queue
- Result
- 2.25 customers
- Measure
- Ws, time in system
- Result
- 30 minutes
MCA 251 Unit 4: Game Theory, Replacement and Simulation
Unit 4 (pages 41 to 50) covers saddle points, dominance, the 2 × 2 mixed-strategy formula and the graphical method for 2 × n games; replacement of ageing machines and of items that fail suddenly (individual vs group policy); and a Monte-Carlo demand simulation. It has 6 solved examples, 10 short answers and 7 long questions.
| Replace after year | Average cost per year (₹) |
|---|---|
| 2 | 1850.00 |
| 3 | 1766.67 (minimum: replace here) |
| 4 | 1775.00 |
Quick look: when to replace a ₹7000 machine, average annual cost by replacement year (from page 46)
- Replace after year
- 2
- Average cost per year (₹)
- 1850.00
- Replace after year
- 3
- Average cost per year (₹)
- 1766.67 (minimum: replace here)
- Replace after year
- 4
- Average cost per year (₹)
- 1775.00
PERT vs CPM and network drawing rules in project scheduling: excerpt from page 36 of the MCA 251 notes
Excerpt from page 36
Basic difference between PERT and CPM
| Basis | PERT | CPM |
|---|---|---|
| Nature | Probabilistic | Deterministic |
| Time estimates | Three (to, tm, tp) | One (single time) |
| Orientation | Event-oriented | Activity-oriented |
| Used for | R&D, new and non-repetitive projects | Construction, repetitive projects |
| Focus | Time | Time and cost (crashing) |
| Critical path | Longest expected-time path | Longest-duration path |
| Slack or float | Slack (for events) | Float (for activities) |
PERT vs CPM (excerpt, page 36)
- Basis
- Nature
- PERT
- Probabilistic
- CPM
- Deterministic
- Basis
- Time estimates
- PERT
- Three (to, tm, tp)
- CPM
- One (single time)
- Basis
- Orientation
- PERT
- Event-oriented
- CPM
- Activity-oriented
- Basis
- Used for
- PERT
- R&D, new and non-repetitive projects
- CPM
- Construction, repetitive projects
- Basis
- Focus
- PERT
- Time
- CPM
- Time and cost (crashing)
- Basis
- Critical path
- PERT
- Longest expected-time path
- CPM
- Longest-duration path
- Basis
- Slack or float
- PERT
- Slack (for events)
- CPM
- Float (for activities)
PERT/CPM network components
- Activity: a task that uses time and resources, drawn as an arrow. The tail is its start and the head is its end.
- Event (node): the start or finish point of activities, drawn as a numbered circle. An event takes no time.
- Dummy activity: a dashed arrow with zero time and zero resources, used only to show a logical dependency or to stop two activities sharing the same start and end nodes.
- Merge event: a node where two or more activities end. Burst event: a node where two or more activities start.
- Path: an unbroken chain of activities from the start node to the end node.
Precedence relationships and rules for drawing a network
- Each activity is shown by one arrow, and only one.
- No two activities can have the same start and end nodes; add a dummy instead.
- An activity starts only when all its predecessors are complete.
- Arrows run left to right; avoid crossing arrows.
- No loops and no dangling activities: every activity except the last must lead to some node.
- One start node and one end node only, numbered so that the tail number is less than the head number (Fulkerson's rule).
Forward and backward pass
In the forward pass, the earliest time of event j is the largest value of Ei + tij over all activities entering j, starting from E1 = 0. In the backward pass, the latest time of event i is the smallest value of Lj − tij over all activities leaving i, and the last event's latest time equals its earliest time. For any activity (i, j): ES = Ei, EF = Ei + t, LF = Lj and LS = Lj − t.
Do the MCA 251 notes cover the full GGSIPU syllabus?
Yes. Every topic block under Units I to IV of the official MCA-251 syllabus has a matching section in the notes, including the Lingo, Excel Solver and TORA work named in Units I and II.
| Official unit | Official topics (shortened) | Where in the notes | Status |
|---|---|---|---|
| Unit I | Introduction to OR; LPP formulation, graphical method, simplex, Big-M, two-phase, duality, sensitivity analysis; LPP in Lingo/Excel Solver/TORA | Pages 4-15 | Covered |
| Unit II | Transportation: NWC, least cost, VAM, MODI, unbalanced, maximisation, degeneracy; assignment: Hungarian method, special cases, travelling salesperson; TP in software | Pages 18-28 | Covered |
| Unit III | Infinite queuing models, single and multiple channels, Erlang service times, applications, limitations; PERT and CPM, network components, time-cost trade-off | Pages 31-39 | Covered |
| Unit IV | Game theory: maximin-minimax, dominance, mixed strategy; replacement of deteriorating and suddenly failing items; simulation methodology and applications | Pages 42-49 | Covered |
MCA 251 official syllabus vs the notes
- Official unit
- Unit I
- Official topics (shortened)
- Introduction to OR; LPP formulation, graphical method, simplex, Big-M, two-phase, duality, sensitivity analysis; LPP in Lingo/Excel Solver/TORA
- Where in the notes
- Pages 4-15
- Status
- Covered
- Official unit
- Unit II
- Official topics (shortened)
- Transportation: NWC, least cost, VAM, MODI, unbalanced, maximisation, degeneracy; assignment: Hungarian method, special cases, travelling salesperson; TP in software
- Where in the notes
- Pages 18-28
- Status
- Covered
- Official unit
- Unit III
- Official topics (shortened)
- Infinite queuing models, single and multiple channels, Erlang service times, applications, limitations; PERT and CPM, network components, time-cost trade-off
- Where in the notes
- Pages 31-39
- Status
- Covered
- Official unit
- Unit IV
- Official topics (shortened)
- Game theory: maximin-minimax, dominance, mixed strategy; replacement of deteriorating and suddenly failing items; simulation methodology and applications
- Where in the notes
- Pages 42-49
- Status
- Covered
Verify the topics yourself in the official MCA syllabus for affiliated institutes, A.S. 2025-26 (PDF, ipu.ac.in); MCA-251 starts on printed page 167.
Course outcome CO6 mentions integer programming, but no unit lists it, so the MCA 251 Operational Research IPU notes name it only among OR tools (page 6). Every subject in GGSIPU notes is mapped to its official PDF the same way.
How to write MCA 251 answers: LPP formulation, simplex tables and interpreted numerical results
Open every LPP with the four formulation steps on page 7, ending with x1, x2 ≥ 0. In simplex, show each table with its Cj − Zj row, mark the entering column (↑) and leaving row (←), and state the stopping rule: for maximisation, all Cj − Zj ≤ 0.
For transportation, use VAM when no method is named (page 20), count occupied cells against m + n − 1 before MODI, and check the saving equals |Δ| × θ (page 22: 18 × 2 = 36). For queuing, confirm ρ < 1, then verify Ls = λWs. End each numerical with one line of meaning, as page 10 does: "s1 = 2 means 2 hours of Plant 1 are unused".
Worked example: chance of finishing a PERT project in 20 weeks (page 38)
- Expected times te = (to + 4tm + tp) / 6 give critical path A-C-F, Te = 18 weeks.
- Add variances of critical activities only: 0.444 + 2.778 + 0.444 = 3.667, so σ = 1.915.
- Z = (20 − 18) / 1.915 = 1.04; the normal table gives 0.8508, about 85%.
- For a negative Z (16 weeks gives −1.04), take 1 − table value: about 15%.
Practise this format on GGSIPU MCA important questions.
MCA 251 exam corner: 40 short-answer and 35 long questions by unit
| Unit | Short answers | Long questions | Solved examples |
|---|---|---|---|
| Unit 1 | 10 | 10 | 6 |
| Unit 2 | 10 | 9 | 5 |
| Unit 3 | 10 | 9 | 5 |
| Unit 4 | 10 | 7 | 6 |
| Total | 40 | 35 | 22 |
MCA 251 notes: questions and solved examples per unit
- Unit
- Unit 1
- Short answers
- 10
- Long questions
- 10
- Solved examples
- 6
- Unit
- Unit 2
- Short answers
- 10
- Long questions
- 9
- Solved examples
- 5
- Unit
- Unit 3
- Short answers
- 10
- Long questions
- 9
- Solved examples
- 5
- Unit
- Unit 4
- Short answers
- 10
- Long questions
- 7
- Solved examples
- 6
- Unit
- Total
- Short answers
- 40
- Long questions
- 35
- Solved examples
- 22
The notes label these "2.5 Marks" and "12.5 Marks". The 2025-26 paper has a 12-mark Q1 and 12-mark unit questions, so use the short answers for Q1 and the long questions, with the solved examples, for the units.
The MCA 251 notes answer 40 short-answer questions and list 35 long questions across 4 units, plus 22 solved numericals.
One question per unit, with answer pointers:
- Unit 1: What is a shadow price? Pointer: the dual variable value: the gain in optimal Z per extra unit of a resource (₹1.5 per Plant 2 hour, page 14).
- Unit 2: Why m + n − 1 basic cells? Pointer: there are m + n constraints, but one is redundant because total supply equals total demand.
- Unit 3: State Little's formula. Pointer: Ls = λWs and Lq = λWq.
- Unit 4: Define a saddle point. Pointer: the cell where maximin equals minimax; its value is the value of the game.
More practice sits in the MCA 251 important questions.
Who writes and checks the MCA 251 Operations Research notes?
- Prateek Talwar, author: Founder, Unnati Educations; Associate Company Secretary (ACS), ICSI, membership no. 58704, since 15 May 2019; B.Com, University of Delhi (School of Open Learning), 2012. Author bio
- Ritik Rawat, subject reviewer: Senior Full Stack Developer, Unnati Educations; B.Tech.
The 2026-27 edition is mapped to the MCA syllabus for affiliated institutes (A.S. 2025-26) and was updated on 14 October 2026 under our editorial policy.
The same team prepares GGSIPU important questions for every course.
MCA 251 and the GGSIPU MCA Sem 3 papers it connects to
Paired lab: none. The full term, with every elective option, is on the GGSIPU MCA syllabus page. Three Sem 3 syllabi reuse MCA 251's methods:
- MCA-201 Design and Analysis of Algorithms lists the assignment problem and travelling salesman problem, solved in Unit 2 (pages 23-27).
- MCA-215 Software Project Management lists critical path analysis and PERT, worked in Unit 3 (pages 35-39).
- MCA-205 Statistics and Data Analytics covers the Poisson and normal distributions behind Unit 3's queues and PERT probabilities.
| Group | Papers |
|---|---|
| Core | MCA-201 Design and Analysis of Algorithms; MCA-203 Artificial Intelligence and Machine Learning |
| Core Elective II | One of MCA-205 to MCA-225 |
| Core Elective III | One of MCA-227 to MCA-247 |
| Open Elective II alternatives | MCA-249 Research Methodology; 253 Cyber Security and Cyber Laws; 255 e-Content Development |
| Labs, project, NUES | MCA-261 DAA Lab; 263 AI and ML Lab; 265 and 267 Elective Labs; 269 Minor Project III; 271 Entrepreneurship Mindset; 273 Professional Proficiency III |
Other MCA Semester 3 papers (A.S. 2025-26 scheme)
- Group
- Core
- Papers
- MCA-201 Design and Analysis of Algorithms; MCA-203 Artificial Intelligence and Machine Learning
- Group
- Core Elective II
- Papers
- One of MCA-205 to MCA-225
- Group
- Core Elective III
- Papers
- One of MCA-227 to MCA-247
- Group
- Open Elective II alternatives
- Papers
- MCA-249 Research Methodology; 253 Cyber Security and Cyber Laws; 255 e-Content Development
- Group
- Labs, project, NUES
- Papers
- MCA-261 DAA Lab; 263 AI and ML Lab; 265 and 267 Elective Labs; 269 Minor Project III; 271 Entrepreneurship Mindset; 273 Professional Proficiency III
MCA 251 Operations Research notes FAQs
How do I get the full MCA 251 notes?
Message us on WhatsApp at 9899436384 with the code MCA 251. The full notes come only as a printed book, not as a PDF, and price and payment details are shared in the chat. Delivery takes 5 to 7 days at most, depending on your location. You can check six real pages first through the View Sample PDF button at the top.
What is the passing mark in MCA 251 Operational Research?
MCA 251 is marked out of 100: 40 internal and 60 end-term. The MCA scheme of examination states pass marks as 40 out of 100 for every paper, which is 40%. For end-term minimums and re-appear rules, the scheme points to Ordinance 11 on ipu.ac.in, linked in the snapshot above; check its current version there.
Is MCA 251 compulsory in MCA Semester 3?
No. MCA 251 is one of four Open Elective II options (MCA-249, 251, 253 and 255), and you pick one, worth 3 credits. Under scheme note 3, an elective chosen by fewer than 20 students may run through a MOOC platform with Programme Co-ordination Committee approval, so confirm the teaching mode with your institute early.
Do the MCA 251 notes show Excel Solver, LINGO and TORA?
Yes. The syllabus names this software in Units I and II. Pages 14-15 set up the product-mix LPP in Excel Solver step by step, give a five-line LINGO model with its output (X1 = 2, X2 = 6, objective 36) and explain TORA. Pages 27-28 do the same for transportation, where LINGO returns the MODI cost of ₹743.
Which MCA 251 unit has the most solved numericals?
Units 1 and 4 have six solved examples each. Unit 1 is the longest block at 14 pages, because one product-mix LPP is carried through graph, simplex, dual and sensitivity. Unit 4 spreads its six across games, replacement and simulation. Units 2 and 3 have five each, though Unit 2's transportation example alone runs over five pages.
Is there a separate MCA 251 lab paper?
No. MCA 251 is a 3-credit theory open elective with no paired practical in the Semester 3 scheme; the Sem 3 labs MCA-261 to MCA-267 belong to algorithms, AI and ML, and the two core electives. Software practice for this paper, in Excel Solver, LINGO and TORA, is built into Units 1 and 2 of the theory notes.
Get the full MCA 251 Operational Research Notes on WhatsApp
The full MCA 251 notes are sold as a printed book only, not as a PDF. They are Operational Research notes in English, with short "simple words mein" Hinglish lines where a step needs a plainer explanation. Price and payment details are shared on WhatsApp.
| Part | Pages | Solved examples |
|---|---|---|
| Cover and index | 1-2 (2) | – |
| Unit 1: Introduction to OR and Linear Programming | 3-16 (14) | 6 |
| Unit 2: Transportation and Assignment Problems | 17-29 (13) | 5 |
| Unit 3: Queuing Theory and PERT/CPM | 30-40 (11) | 5 |
| Unit 4: Game Theory, Replacement and Simulation | 41-50 (10) | 6 |
| Total | 50 | 22 |
MCA 251 notes book: pages per unit
- Part
- Cover and index
- Pages
- 1-2 (2)
- Solved examples
- –
- Pages
- 3-16 (14)
- Solved examples
- 6
- Pages
- 17-29 (13)
- Solved examples
- 5
- Pages
- 30-40 (11)
- Solved examples
- 5
- Pages
- 41-50 (10)
- Solved examples
- 6
- Part
- Total
- Pages
- 50
- Solved examples
- 22
Delivery: within 5 to 7 days at most; the exact time may vary with your location.
Get the MCA 251 Operational Research Notes book on WhatsApp (4 units, 50 pages)
Unnati Educations is an independent education provider based in Vikaspuri, Delhi. We are not affiliated with, authorised by, or endorsed by Guru Gobind Singh Indraprastha University (GGSIPU). 'GGSIPU' and 'IPU' are used only to describe the syllabus our notes follow. For official information, visit ipu.ac.in.
Ready for the full notes?
Discuss coverage and delivery with our team.