a. Define Operations Research and mention its scope.

Definition: Operations Research (OR) ek scientific aur analytical approach hai jo decision-making ke liye mathematical models ka use karti hai. Iska maqsad limited resources ka optimal use karna hai.

Scope: Iska use Production (Product mix), Marketing (Advertising budget), Finance (Capital budgeting), aur Logistics (Routing) mein hota hai.

b. What is the North West Corner Method in transportation problems?

North West Corner Method (NWCM) transportation problem ka Initial Basic Feasible Solution (IBFS) nikalne ka sabse simple tarika hai. Isme hum allocation table ke ekdum **top-left (North-West)** corner se shuru karte hain aur demand-supply ko satisfy karte huye aage badhte hain.

c. State the Hungarian Method and its use in assignment problems.

Hungarian Method: Ye ek combinatorial optimization algorithm hai jo assignment problems ko solve karne ke liye use hota hai. Ye total cost ya time ko minimize karne mein help karta hai.

Use: Iska use jobs ko machines par assign karne, ya employees ko tasks assign karne mein hota hai taki minimum effort mein kaam pura ho.

d. Discuss various decision making environments.

Decision making environments teen tarah ke hote hain:

• Under Certainty: Jab har alternative ka outcome pata ho.
• Under Risk: Jab outcomes ki probabilities (sambhvana) pata hon.
• Under Uncertainty: Jab future outcomes ke bare mein koi data available na ho.

e. Explain the concept of sequencing in operations research.

Sequencing ka matlab hai 'n' jobs ko 'm' machines par ek aise order mein process karna ki **Total Elapsed Time** (poora kaam khatam hone ka time) aur machines ka **Idle Time** minimum ho jaye. Isme Johnson’s Rule ka use aksar kiya jata hai.

f. What is replacement analysis? Give an example of its application.

Replacement Analysis: Ye wo process hai jisme decide kiya jata hai ki kisi purani machine ya equipment ko kab badalna (replace) behtar hoga. Ye maintenance cost aur purchase cost ke balance ko dekhta hai.

Example: Ek delivery truck ko 5 saal baad replace karna kyunki uski repair ka kharcha naye truck ki EMI se zyada ho raha hai.

g. List two differences between CPM and PERT in project management.

Q2(a) Role of Operations Research in Inventory Management & Decision-Making

Operations Research (OR) inventory management mein ek backbone ki tarah kaam karta hai. Ye decision-makers ko balance banane mein help karta hai taki na toh stock khatam ho (Stock-out) aur na hi faltu stock jama ho (Overstocking).

Key Roles:

• Economic Order Quantity (EOQ): OR model se ye pata chalta hai ki ek baar mein kitna saman mangwaya jaye taki ordering aur carrying cost dono minimum ho.
• Safety Stock Level: Uncertainty ko handle karne ke liye kitna buffer stock rakhna hai, ye OR ke probabilistic models batate hain.
• Lead Time Analysis: Order dene aur saman pahunchne ke beech ke gap ko manage karna.
• ABC Analysis: Inventory ko unki value aur importance ke hisaab se prioritize karna (Always Better Control).

Conclusion: OR techniques se inventory holding cost 20-30% tak kam ki ja sakti hai, jo direct profit mein contribute karti hai.

Q2(b) Discuss Simplex Method with a Suitable Example

Simplex Method ek iterative procedure hai jo complex Linear Programming Problems (LPP) ko solve karne ke liye use hota hai jahan variable do se zyada hon.

Steps in Simplex Method:

1. Standard Form: Constraints ko equations mein badalne ke liye Slack ya Surplus variables add karein.
2. Initial Table: Ek basic feasible solution (BFS) ke saath Simplex Table banayein.
3. Optimality Test: Cj - Zj values calculate karein. Agar saari values ≤ 0 hain (Maximization ke liye), toh solution optimal hai.
4. Pivot Element: Agar optimal nahi hai, toh entering variable (Key Column) aur leaving variable (Key Row) select karke naya table banayein.

Example:

Maximize $Z = 3x_1 + 5x_2$ subject to $2x_1 + x_2 \le 4$, $x_1 + 2x_2 \le 5$.
Isme hum $s_1$ aur $s_2$ slack variables add karke table banayenge jab tak $x_1$ aur $x_2$ ki aisi values na mil jayein jo $Z$ ko maximum kar dein.

Q2(c) Steps Involved in Solving a Transportation Problem

Transportation problem ka maqsad goods ko source (factory) se destination (warehouse) tak minimum cost par pahunchana hai.

Three Major Phases:

• 1. Mathematical Formulation: Sabse pehle objective function (Min Z) aur constraints (Supply & Demand) ko define karna.
• 2. Initial Basic Feasible Solution (IBFS): Iske liye niche diye gaye methods use hote hain:
  • North West Corner Rule (Simple par costly).
  • Least Cost Method (Cost ko dhyan mein rakhta hai).
  • Vogel’s Approximation Method (VAM): Ye sabse best hai kyunki ye penalties consider karta hai.
• 3. Optimality Test: Check karna ki kya cost aur kam ho sakti hai. Iske liye MODI (Modified Distribution) Method ya Stepping Stone method use hota hai.

Q2(d) Sequencing Problem for n Jobs and 2 Machines

Sequencing ka goal hai ki 'n' jobs ko Machine A aur B se aise guzara jaye ki poora kaam (Total Elapsed Time) jaldi khatam ho.

Johnson’s Algorithm (Steps):

1. Dono machines (A aur B) ke saare processing times list karein.
2. Sabse chhota (minimum) time dhoondhein.
3. Agar min time Machine A par hai, toh us job ko sabse pehle rakhein.
4. Agar min time Machine B par hai, toh us job ko sabse aakhri mein rakhein.
5. Us job ko list se hata dein aur process repeat karein jab tak sequence poora na ho jaye.

Example: Agar Job 1 ko A par 2 ghante lagte hain aur B par 5, toh hum Job 1 ko pehle karenge kyunki min time A par hai.

Q2(e) Explain the GANTT Chart

Gantt Chart ek visual project management tool hai jo Henry Gantt ne develop kiya tha. Ye ek horizontal bar chart hai jo project ke schedule ko dikhata hai.

Key Features:

• Timeline: X-axis par time (din, hafte) dikhaya jata hai.
• Tasks: Y-axis par saari activities list hoti hain.
• Bars: Har task ki duration ek bar se dikhayi jati hai—jitni lambi bar, utna zyada time.
• Dependencies: Ye dikhata hai ki kaunsa kaam khatam hone ke baad hi agla kaam shuru ho sakta hai.

Use: Iska use resources allocate karne aur project ki progress track karne mein hota hai taaki deadlines miss na hon.

Q3(a) Discuss the statement that OR is Science and an Art both?

Operations Research (OR) ko Science aur Art dono mana jata hai kyunki ye logic aur creativity ka ek anokha sangam hai.

• As a Science: Ye scientific methods par based hai. Isme structured models, mathematical algorithms (Simplex, Hungarian), aur data analysis ka use hota hai. Iska approach logical aur predictable hota hai.
• As an Art: Sirf formula lagana kafi nahi hota. Real-world problems ko mathematical form mein badalna (formulation) ek art hai. Isme manager ka experience aur intuition kaam aata hai ki kaunsa model kahan fit baithega.

Conclusion: OR ka "Science" part humein tools deta hai, jabki "Art" part humein un tools ko sahi jagah apply karna sikhata hai.

Q3(b) Discuss various models of Operations Research.

OR mein situation ke hisaab se alag-alag models use hote hain:

• Iconic Models: Ye physical look-alike hote hain (jaise car ka miniature model).
• Mathematical (Symbolic) Models: Problems ko variables aur equations mein dikhana (jaise LPP).
• Static Models: Jo time ke saath badalte nahi (Deterministic models).
• Dynamic Models: Jo waqt ke saath change hote hain (Inventory models).
• Simulation Models: Real-world scenarios ko computer par copy karna.

Q4(a) Graphical Method: Maximize Z = 3X + 4Y

Constraints: X + 2Y ≤ 30 | 2X + 7Y ≥ 14 | X, Y ≥ 0

Step 1: Intercepts Nikalna

• Line 1 (X + 2Y = 30): Agar X=0 toh Y=15; Agar Y=0 toh X=30. Points: (0, 15) and (30, 0).
• Line 2 (2X + 7Y = 14): Agar X=0 toh Y=2; Agar Y=0 toh X=7. Points: (0, 2) and (7, 0).

Step 2: Feasible Region

Line 1 ke niche (≤) aur Line 2 ke upar (≥) ka area feasible region hoga.

Step 3: Corner Points Value

• At (7, 0): Z = 3(7) + 4(0) = 21
• At (30, 0): Z = 3(30) + 4(0) = 90
• At (0, 15): Z = 3(0) + 4(15) = 60
• At (0, 2): Z = 3(0) + 4(2) = 8

Optimal Result: Maximum Value of Z is 90 at (30, 0).

Q5(a) Assignment Problem: Minimize Total Cost

Is matrix ko Hungarian Method se solve karte hain:

Optimal Assignment Logic: Row aur Column reduction ke baad hum zeroes ko assign karenge. Is matrix mein minimum values (Bold) par hi assignment ban rahi hai.

Minimum Cost: 5 (W1-J4) + 1 (W2-J5) + 5 (W3-J1) + 3 (W4-J2) + 3 (W5-J3) = 17.

Q5(b) Differentiate between Assignment and Transportation Problems

Q6(a) Processing n jobs through two machines & Johnson's Algorithm

Jab humein 'n' number of jobs ko do machines (M1 aur M2) se guzarna hota hai, toh hum Johnson’s Rule ka use karte hain taaki idle time minimum ho.

The Algorithm:

1. Saari jobs ke processing times ko do columns (Machine A aur B) mein likhein.
2. Puri list mein se minimum processing time dhoondhein.
3. Agar minimum time Machine A ka hai, toh us job ko sequence mein pehle (left to right) rakhein.
4. Agar minimum time Machine B ka hai, toh us job ko sequence mein aakhri (right to left) rakhein.
5. Us job ko list se kaat dein aur baki jobs ke liye yahi process repeat karein.

Purpose: Isse total elapsed time (makespan) minimize hota hai aur resources ka efficient use hota hai.

Q6(b) Key Elements of a Queuing System & LIFO Discipline

Ek queuing system ke main components niche diye gaye hain:

• Input Source: Customers ki sankhya (Finite ya Infinite).
• Arrival Process: Customers kis speed se aa rahe hain (aksar Poisson distribution).
• Service Mechanism: Service dene ka tarika (Single channel ya Multi-channel).
• Queue Capacity: Line mein kitne log khade ho sakte hain.

What is LIFO (Last-In-First-Out)?

Is discipline mein jo customer sabse aakhri mein aata hai, use sabse pehle service milti hai.
Example: Ek elevator (lift) mein jo log sabse baad mein ghuste hain, wo gate ke paas hone ki wajah se sabse pehle naye floor par nikalte hain.

Q7(a) Replacement Policy for Equipment based on Optimal Time

Equipment replacement policy tab develop ki jati hai jab purani machine ki maintenance cost naye machine ke kharche se zyada hone lage.

Optimal Replacement Time Justification:

• Hum har saal ki Average Annual Cost calculate karte hain.
• Is cost mein (Capital Cost - Resale Value) + Cumulative Operating Cost shamil hoti hai.
• Policy: Jab tak average annual cost girti hai, machine ko chalne dein. Jis saal average cost badhna shuru ho jaye, wahi uska optimal replacement time hai.

Q7(b) Constructing a Network Diagram & Critical Path (CPM)

CPM (Critical Path Method) ka use project timeline ko visualize karne ke liye kiya jata hai.

Steps to Construct:

1. Nodes & Arrows: Har activity ko arrow se aur milestone ko circle (node) se dikhayein.
2. Precedence: Di gayi table ke hisaab se dekhein ki 'A' ke baad 'B' aayega ya 'E' ke baad 'F'.
3. Forward Pass: Earliest Start Time (EST) calculate karein.
4. Backward Pass: Latest Finish Time (LFT) calculate karein.

Critical Path: Wo lamba raasta jahan har activity ka Float Zero hota hai. Agar is path par koi bhi delay hua, toh pura project late ho jayega.