Tutorial 8 Exercises

Question 1

VoltHub is a consumer electronics retailer operating in Hamburg. At one of its city-center stores, the expected demand for wireless earbuds is 120 units per month. Each unit costs €50, and the fixed ordering cost is €60 per order. The company applies an annual holding cost rate of 15%. The forecasting MSE for monthly demand is 36. The required service level is 95%.

  1. (Reorder Point Policy) Determine the optimal order quantity (EOQ), safety stock, and reorder point assuming a deterministic lead time of 2 weeks.

  2. (Periodic Review Policy) Determine the optimal review period 𝑇, the maximum inventory level 𝑆, and the associated safety stock, assuming the lead time is stochastic with a mean of 0.5 months and a variance of 0.25 months².

Question 2: EOQ and Reorder Point (Deterministic Case)

A retailer faces an annual demand of \(D = 12{,}000\) units. The ordering cost is \(K = \$100\) per order, and the holding cost is \(h = \$0.50\) per unit per year. The lead time is 8 days, and the store operates 300 days per year.

  1. Compute the Economic Order Quantity (EOQ).
  2. Determine the number of orders per year.
  3. Calculate the reorder point.

Question 3: Reorder Point with Demand Uncertainty

A product has an average weekly demand of 150 units with a standard deviation of 20 units. The lead time is 4 weeks. The desired service level is 95%.

  1. Compute the standard deviation of demand during lead time.
  2. Determine the safety stock.
  3. Calculate the reorder point.

Question 4: Variable Lead Time

Daily demand is constant at 50 units. Lead time is normally distributed with a mean of 6 days and a standard deviation of 2 days. The desired service level is 90%.

  1. Determine the safety stock.
  2. Compute the reorder point.

Question 5: EOQ with Total Cost

A company has annual demand \(D = 20{,}000\) units, ordering cost \(K = \$200\), and holding cost \(h = \$1\) per unit per year.

  1. Compute the EOQ.
  2. Calculate the total annual inventory cost at EOQ.

Question 6: Periodic Review System

Demand is 30 units per week. The review period is \(P = 4\) weeks, and lead time is \(L = 2\) weeks.

  1. Compute the order-up-to level (\(TIL\)) assuming no uncertainty.

  2. Now assume demand has a standard deviation of 6 units per week and the service level is 95%. - Compute the safety stock.

    • Determine the new \(TIL\).

Question 7: Comparison of \(Q\) and \(P\) Systems

A product has an EOQ of 120 units and an average demand of 40 units per week. Lead time is 2 weeks and the service level is 95%. Weekly demand standard deviation is 5 units.

  1. Determine the review period \(P\).
  2. Compute the safety stock.
  3. Calculate the order-up-to level (\(TIL\)).

Question 8: Comprehensive Inventory Management Problem

An owner of a store has estimated an annual demand of product A to be 10,000 units, an annual holding cost of $0.5 per unit, and an ordering cost of $50. If his store is open 200 days per year:

  1. Find the optimal order size, number of orders per year, the cycle length, total annual inventory cost and reorder point if lead time is 5 days.
  2. For demand that is normally distributed with an average daily demand of 55 units and a standard deviation of 10 units per day, find the reorder point and safety stock if the manager wants a service level of 95%, assuming a lead time of 5 days.
  3. If lead time is normally distributed with a mean of 5 days and a standard deviation of 2 days, find the reorder point and safety stock corresponding to a 90% service level.
  4. If demand is normally distributed with an average daily demand of 55 units and a standard deviation of 10 units per day, and lead time is also normally distributed with a mean of 5 days and a standard deviation of 2 days, find the reorder point and safety stock corresponding to an 84.1% service level.
  5. For periodic review policy and using information from (d), determine the periodicity \(T\) of the review period, reorder point (R), safety stock (SS), and the maximum inventory level associated with safety stock corresponding to a 97.7% service level.

Question 9: Quantity Discount

A warehouse manages inventory for a high-demand electronic component. Demand is uncertain and follows a normal distribution.

  • The following information is known: demand rate \(d = 200\) units/week, \(\sigma = 40\) units/week, and lead time \(L = 1.5\) weeks.
  • The company targets a 97.5% service level.
  1. Assume the warehouse uses a Continuous Review System \((Q,R)\).
    • Compute the mean demand during lead time.
    • Compute the standard deviation of demand during lead time.
    • Determine the safety stock.
    • Determine the reorder point \((ROP)\).
  2. Now assume the warehouse uses a Periodic Review System \((P\text{-system})\) \(P = 4 \text{ weeks}\).
    • Compute the mean demand during the protection period.
    • Compute the standard deviation of demand during the protection period.
    • Determine the safety stock.
    • Determine the order-up-to level \((TIL)\).

Question 10: Comprehensive Inventory Management Problem with Quantity Discount

A pharmacy manages its inventory of a widely used over-the-counter medication using a \((Q,R)\) system. Historical records show that the weekly demand is approximately normally distributed, with a mean of 80 units and a standard deviation of 15 units. The replenishment lead time is 6 weeks. Each unit costs $10. Any unmet demand is backordered, and the shortage cost is estimated at $20 per unit due to customer dissatisfaction and administrative handling. The fixed ordering cost is $25 per order, and the annual holding cost rate is 20% of the unit cost.

  1. Determine the optimal order quantity \(Q\) and reorder point \(R\).
  2. Compute the expected safety stock just before an order arrives.
  3. Using your results from part (a), explain the roles of \(Q\) and \(R\) in the inventory system, and show how \(R\) is related to demand during lead time and safety stock.