Review & Practical Applications of Operations Research

TAKEAWAYS:

• Operations Research bridges theory (e.g., LPP, game theory) with practical problems like NEPSE stock portfolio optimization or Pathao driver route planning.
• Khalti’s payment processing uses queuing theory to manage transaction queues, while Daraz’s inventory models minimize stockouts.
• NTC’s network routing applies assignment problems, and Ncell’s call-center scheduling relies on queuing models.
• Banks’ loan approvals use decision trees, and YouTube’s ad targeting leverages game theory for bid optimization.
• Compare models (e.g., PERT vs. CPM) and identify their strengths/weaknesses in project timelines.
• Always map real problems to the correct OR tool (e.g., linear programming for resource allocation, queuing for service delays).

1. Recap of Core OR Models

Operations Research (OR) solves complex decision-making problems using mathematical models. Below is a comparison table of the key techniques covered in the course, with their applications in Nepal and globally.

classDiagram
    class ORModel {
        <<abstract>>
        +name: String
        +description: String
        +applications: String[]
    }

    class LinearProgramming {
        +name: "Linear Programming (LPP)"
        +description: Optimizes linear objectives under constraints
        +applications: ["NEPSE stock allocation", "Daraz inventory", "Bank loan approvals"]
    }

    class TransportationProblem {
        +name: "Transportation Problem"
        +description: Minimizes cost of distributing goods
        +applications: ["NTC network routing", "Supply chain logistics"]
    }

    class AssignmentProblem {
        +name: "Assignment Problem"
        +description: Assigns tasks/resources optimally
        +applications: ["Pathao driver allocation", "Call-center staffing"]
    }

    class GameTheory {
        +name: "Game Theory"
        +description: Models strategic decisions
        +applications: ["Advertising bids (Google/YouTube)", "Nepalese election strategies"]
    }

    class PERTCPM {
        +name: "PERT/CPM"
        +description: Manages project timelines
        +applications: ["Ncell network expansion", "Hydropower dam construction"]
    }

    class QueuingTheory {
        +name: "Queuing Theory"
        +description: Models waiting lines
        +applications: ["Khalti payment queues", "NTC customer service"]
    }

    class InventoryManagement {
        +name: "Inventory Management"
        +description: Balances stock and demand
        +applications: ["Daraz warehouse optimization", "Nepalese spice imports"]
    }

    ORModel <|-- LinearProgramming
    ORModel <|-- TransportationProblem
    ORModel <|-- AssignmentProblem
    ORModel <|-- GameTheory
    ORModel <|-- PERTCPM
    ORModel <|-- QueuingTheory
    ORModel <|-- InventoryManagement

2. Practical Applications in Nepal and Globally

A. Financial and Business Applications

1. NEPSE Stock Portfolio Optimization (Linear Programming)

Real-world use: NEPSE (Nepal Stock Exchange) uses linear programming to optimize stock portfolios for investors, balancing risk and return. How it works:

• Objective: Maximize profit or minimize risk.
• Constraints:
  • Budget limits (e.g., ₹500,000).
  • Sector allocation (e.g., 30% in banking, 20% in tourism).
  • Risk tolerance (e.g., no more than 40% in volatile stocks).

Worked Example: A investor has ₹500,000 to invest in Nepal Bank (NB), Standard Chartered Bank (SCB), and Himalayan Bank (HB). The expected returns and constraints are:

• Returns: NB = ₹12%, SCB = ₹15%, HB = ₹10%.
• Constraints:
  • Max ₹200,000 in NB.
  • Max ₹150,000 in SCB.
  • At least ₹100,000 in HB.

Graphical Solution: We plot the constraints and find the feasible region. The optimal solution lies at the intersection of SCB and HB constraints.

Solution:

• Invest ₹200,000 in NB, ₹150,000 in SCB, and ₹150,000 in HB.
• Expected profit: ₹12% of 200k + ₹15% of 150k + ₹10% of 150k = ₹60,000.

Advantages:

• Maximizes returns under constraints.
• Reduces risk by diversifying investments.

Disadvantages:

• Assumes linear returns (real markets are nonlinear).
• Ignores market volatility.

2. Khalti’s Payment Queue Management (Queuing Theory)

Real-world use: Khalti processes thousands of transactions per second, using queuing theory to manage delays. How it works:

• Arrival rate (λ): Transactions per minute (e.g., 500/min).
• Service rate (μ): Transactions processed per minute (e.g., 600/min).
• Queue length (L): Average number of pending transactions.
• Waiting time (W): Average delay before processing.

Worked Example: Khalti’s server handles 500 transactions/minute, but the system processes 600/minute. Calculate:

1. Utilization (ρ): ρ = λ/μ = 500/600 = 0.833.
2. Average queue length (L): L = ρ² / (1 - ρ) = (0.833)² / (1 - 0.833) ≈ 4.76 transactions.
3. Average waiting time (W): W = L / λ ≈ 4.76 / 500 ≈ 0.0095 minutes (≈0.57 seconds).

Visualization:

Advantages:

• Predicts bottlenecks (e.g., peak hours).
• Optimizes server resources.

Disadvantages:

• Assumes Poisson arrivals (real-world traffic is bursty).
• Ignores network latency.

B. Logistics and Transportation

1. NTC’s Network Routing (Assignment Problem)

Real-world use: NTC (Nepal Telecom) uses assignment problems to route calls efficiently across towers. How it works:

• Tasks: Calls from users.
• Resources: Cell towers.
• Goal: Assign calls to towers to minimize cost/delay.

Worked Example: NTC has 3 towers (A, B, C) and 3 users (U1, U2, U3). The cost matrix (in ₹) is:

Solution using the Assignment Algorithm:

1. Subtract row minima:

   • Row A: Subtract 5 → [0, 2, 1]
   • Row B: Subtract 4 → [4, 0, 5]
   • Row C: Subtract 5 → [1, 0, 2]

2. Subtract column minima:

   • Column 1: Subtract 0 → [0, 4, 1]
   • Column 2: Subtract 0 → [2, 0, 0]
   • Column 3: Subtract 1 → [1, 5, 1]

3. Assign zeros (optimal assignments):

   • U2 → B (cost 4)
   • U3 → A (cost 6)
   • U1 → C (cost 6)
   • Total cost: 4 + 6 + 6 = ₹16.

Visualization:

flowchart TD
    A["Tower A"] -->|"U3: ₹6"| U3["User 3"]
    B["Tower B"] -->|"U2: ₹4"| U2["User 2"]
    C["Tower C"] -->|"U1: ₹6"| U1["User 1"]

Advantages:

• Minimizes call routing costs.
• Reduces network congestion.

Disadvantages:

• Assumes fixed costs (real-world costs vary by time).

2. Daraz’s Inventory Management (Inventory Models)

Real-world use: Daraz uses Economic Order Quantity (EOQ) to balance stock and ordering costs. How it works:

• Demand (D): Units sold per year (e.g., 10,000 units).
• Order cost (S): Cost per order (e.g., ₹200).
• Holding cost (H): Cost per unit per year (e.g., ₹5).
• EOQ formula: $Q=\sqrt{\frac{2DS}{H}}$.

Worked Example: Daraz sells 10,000 units/year of a product with:

• Order cost (S) = ₹200.
• Holding cost (H) = ₹5/unit/year.

Calculation: $$Q=\sqrt{\frac{2\times 10,000\times 200}{5}}=\sqrt{80,000}\approx 282.84$$ Order every 283 units.

Total Cost (TC): $$TC=\frac{D}{Q}\times S+\frac{Q}{2}\times H$$ $$TC=\frac{10,000}{283}\times 200+\frac{283}{2}\times 5\approx ₹7,060$$

Visualization:

Advantages:

• Reduces holding and ordering costs.
• Prevents stockouts.

Disadvantages:

• Assumes constant demand (real demand fluctuates).
• Ignores lead time.

C. Project Management

1. Ncell’s Network Expansion (PERT/CPM)

Real-world use: Ncell uses PERT/CPM to schedule 5G tower installations. How it works:

• Activities: Tasks (e.g., site preparation, wiring).
• Durations: Time per task (weeks).
• Critical Path: Longest path determines project duration.

Worked Example: Ncell’s project has activities:

Network Diagram:

graph TD
    A["Site Prep (3w)"] --> B["Wiring (2w)"]
    A --> C["Tower Install (4w)"]
    B --> D["Testing (1w)"]
    C --> D

Critical Path:

• A → C → D (3 + 4 + 1 = 8 weeks).

Advantages:

• Identifies delays early.
• Optimizes resource allocation.

Disadvantages:

• Assumes activity durations are fixed (real-world delays occur).

3. Game Theory in Advertising (YouTube/Google)

Real-world use: YouTube uses game theory to optimize ad bidding. How it works:

• Players: Advertisers (e.g., Coca-Cola, Pepsi).
• Strategy: Bid amount for ad slots.
• Payoff: Higher bids win visibility.

Worked Example: Two advertisers (A and B) bid for a slot:

• If A bids ₹100 and B bids ₹80, A wins.
• Payoff matrix:

Nash Equilibrium: Both bid ₹100 (no incentive to deviate).

Visualization:

Advantages:

• Predicts optimal bidding strategies.
• Maximizes ad revenue.

Disadvantages:

• Assumes rational players (real-world bids are emotional).

4. Queuing in Kathmandu Traffic (Real-World Trace)

Real-world use: Kathmandu’s traffic jams can be modeled using M/M/1 queues (single-server, Poisson arrivals). How it works:

• Arrival rate (λ): Vehicles/minute (e.g., 10).
• Service rate (μ): Vehicles cleared/minute (e.g., 8).
• Utilization (ρ): λ/μ = 10/8 = 1.25 (unstable!).

Problem: ρ > 1 → infinite queue (gridlock). Solution: Increase μ (e.g., more lanes) or reduce λ (e.g., congestion pricing).

Visualization:

5. Comparing OR Models

6. Step-by-Step Problem-Solving Approach

1. Identify the Problem Type:

   • Is it optimization (LPP, assignment) or decision-making (game theory)?
   • Is it a project (PERT/CPM) or waiting line (queuing)?

2. Formulate the Model:

   • Write constraints/objective functions.
   • Assign variables (e.g., $x_1$ = investment in NB).

3. Solve Mathematically:

   • Use graphical method (LPP), assignment algorithm, or PERT diagrams.

4. Validate with Real Data:

   • Plug in Nepalese/NEPSE numbers (e.g., stock returns, transaction rates).

5. Compare Alternatives:

   • Check if another model (e.g., CPM vs. PERT) fits better.

6. Optimize:

   • Adjust parameters (e.g., increase μ in queuing to reduce delays).

In the Real World

1. NEPSE Stock Trading:

   • Uses linear programming to allocate ₹500,000 across NB, SCB, and HB for maximum returns (as shown in the worked example).
   • Why it matters: Helps investors avoid risky single-stock bets.

2. Khalti’s Payment Delays:

   • Applies queuing theory to ensure transactions are processed in <1 second during peak hours.
   • Why it matters: Prevents customer frustration and chargebacks.

3. Pathao’s Driver Assignment:

   • Solves assignment problems to match drivers to nearby riders, reducing empty trips.
   • Why it matters: Cuts fuel costs and increases earnings for drivers.

4. Ncell’s 5G Rollout:

   • Uses PERT/CPM to schedule tower installations without delays.
   • Why it matters: Ensures on-time 5G launch in Kathmandu.

5. Daraz’s Warehouse Stock:

   • Implements EOQ models to order inventory every 283 units, balancing costs.
   • Why it matters: Avoids overstocking (wasted space) or stockouts (lost sales).

6. YouTube Ad Auctions:

   • Relies on game theory to set bids that win ad slots without overpaying.
   • Why it matters: Keeps ad costs competitive for businesses.

Exam Tip

1. Always Map to Real Scenarios:

   • If the question is about NEPSE, link it to LPP.
   • If it’s about Khalti, use queuing theory.
   • Example: "A bank uses OR to approve loans. Which model?" → Decision trees or LPP.

2. Show Worked Examples with Numbers:

   • For LPP, plot constraints and shade the feasible region.
   • For queuing, calculate ρ, L, and W with given λ and μ.
   • Example: "Calculate EOQ for Daraz with D=10,000, S=₹200, H=₹5." → Use the formula and graph.

3. Compare Models:

   • The exam may ask: "Why is PERT better than CPM for uncertain projects?"
   • Answer: PERT accounts for probabilistic activity times, while CPM assumes fixed durations.

4. Highlight Assumptions:

   • Always state: "Assuming Poisson arrivals" (queuing) or "Assuming linear returns" (LPP).
   • Example: "This queuing model assumes λ=500/min, but real-world traffic is bursty."

5. Use Diagrams:

   • Draw network diagrams for PERT/CPM.
   • Plot payoff matrices for game theory.
   • Sketch feasible regions for LPP.

6. Time Management:

   • Spend 30% of time understanding the problem.
   • 40% solving (calculations, diagrams).
   • 30% verifying (check assumptions, units, and realism).

Final Note: OR is about turning abstract math into real-world decisions. Always tie your answers to Nepalese examples (NEPSE, Khalti, Pathao) or global ones (YouTube, Amazon) to score full marks. Show your work visually—graphs, diagrams, and tables carry extra credit!