Inventory Models & Waiting Line Analysis: Costs, Queues & Service Systems

Core Concepts: Inventory Management

1. What is Inventory?

Inventory refers to raw materials, work-in-progress items, and finished goods held by a business to meet demand while minimizing costs. Poor inventory management leads to:

• Stockouts (lost sales, customer dissatisfaction).
• Excess inventory (high holding costs, waste).

2. Types of Inventory Costs

Every inventory decision balances three key costs:

• Ordering Cost (S): Fixed cost per order (e.g., placing an order with Daraz).
• Holding Cost (H): Variable cost per unit per time (storage, insurance, obsolescence).
• Stockout Cost (P): Cost of running out (lost profit, emergency orders).

Worked Example 1: Cost-Volume Tradeoff at a Kathmandu Grocery A shop orders 50 kg of rice/month. Annual demand = 600 kg, ordering cost = Rs. 200/order, holding cost = 10% of rice price (Rs. 50/kg). Find: Optimal order quantity (EOQ) to minimize total cost.

EOQ = \sqrt{\frac{2DS}{H}} = \sqrt{\frac{2 \times 600 \times 200}{50}} = \sqrt{4800} \approx 69.3 \text{ kg}

Graph: Interpretation:

• Order 69 kg/6 times/year (not 50 kg/12 times) to save Rs. 700/year.
• Real-world tie: Like how Nepal Food Hub optimizes bulk orders to reduce storage costs.

3. Inventory Models

Worked Example 2: Reorder Point for Ncell’s SIM Cards

• Daily demand (d): 500 SIMs.
• Lead time (L): 3 days.
• Safety stock (SS): 200 SIMs (to cover delays). ROP = $500\times 3+200=1700$ SIMs. Graph:

Core Concepts: Waiting Line (Queuing) Models

Queuing theory studies waiting lines to optimize service systems (e.g., banks, hospitals, call centers).

1. Key Terms

classDiagram
    class ArrivalProcess {
        +λ = arrival rate (customers/hour)
        +Poisson: λt = probability of t arrivals
    }
    class ServiceProcess {
        +μ = service rate (customers/hour)
        +Exponential: 1/μ = avg service time
    }
    class QueueSystem {
        +L = avg queue length
        +W = avg waiting time
        +P₀ = probability of empty system
    }
    ArrivalProcess --> ServiceProcess : λ vs. μ
    ServiceProcess --> QueueSystem : Determines L, W, P₀

2. M/M/1 Model (Single Server)

• Assumptions:
  • Arrivals: Poisson process (λ).
  • Service times: Exponential (μ).
  • Single server (e.g., a Khalti customer service agent).
• Key Metrics:

  L = \frac{\lambda}{\mu - \lambda}, \quad W = \frac{L}{\lambda}, \quad P_0 = 1 - \frac{\lambda}{\mu}
  

Worked Example 3: Khalti’s Chat Support Queue

• λ: 12 customers/hour.
• μ: 15 customers/hour. Find: Avg. waiting time (W) and queue length (L).

L = \frac{12}{15 - 12} = 3 \text{ customers}
W = \frac{3}{12} = 0.25 \text{ hours} = 15 \text{ minutes}
P_0 = 1 - \frac{12}{15} = 0.2 \text{ (20% idle time)}

Graph:

3. M/M/c Model (Multiple Servers)

• Example: NTC’s toll booths (3 lanes, λ = 20 cars/hour, μ = 10 cars/hour/lane).
• Formula (Erlang C):

  L_q = \frac{P_0 (\lambda/\mu)^c \lambda}{c! (c\mu - \lambda)^2} \times P_w
  

  (Use tables/software for complex cases.)

Comparison Table:

## In the Real World

1. eSewa’s Payment Queue (M/M/1)

   • During peak hours (λ = 30 transactions/min), eSewa’s server handles 35 transactions/min (μ).
   • Problem: If λ > μ, users face long waits (seen in Dashain/Tihar seasons).
   • Solution: Add more servers (parallel payment processors) to reduce W.

2. Daraz’s Warehouse (EOQ + ROP)

   • Daraz uses EOQ to order bulk products (e.g., 500 mobile phones) from suppliers.
   • ROP triggers reorders when stock hits 100 units (safety stock = 50).
   • Cost saved: Rs. 200,000/year in holding costs.

3. NTC’s Network Outages (Queuing + Inventory)

   • NTC maintains spare routers (inventory) to fix outages quickly.
   • Queuing: Customers calling support (λ = 50/hour) are routed to 3 agents (μ = 20/hour/agent).
   • Goal: Keep $W<5$ minutes to reduce complaints.

## Exam Tip

1. Inventory Questions:

   • Always draw the cost graph (U-shaped) and mark EOQ.
   • For ROP, state assumptions (e.g., "demand is normal, lead time fixed").
   • Common pitfall: Forgetting safety stock in ROP calculations.

2. Queuing Questions:

   • Label all variables: Clearly define λ, μ, L, W, P₀.
   • Units matter: If λ is "patients/day," ensure μ matches (e.g., "patients/day").
   • Real-world twist: Exams often give arrival/service rates in minutes/hours—convert to consistent units.
   • Shortcut: For M/M/1, memorize:

     L = \frac{\lambda}{\mu - \lambda}, \quad W = \frac{1}{\mu - \lambda}
     

3. Diagrams:

   • Inventory: Sketch a sawtooth graph (inventory level over time).
   • Queuing: Draw a queue system with arrivals (λ) and service (μ) arrows.

Final Note: This unit tests both theory and application. Practice:

• Calculating EOQ/ROP for Nepali businesses (e.g., a local shop’s sugar inventory).
• Solving queuing problems for service systems (e.g., Pathao’s driver wait times). Visualize every step—examiners reward clear, labeled diagrams!