Question 2. Consider an item whose inventory is controlled by a contimuous revie

Question

Question 2. Consider an item whose inventory is controlled by a contimuous review policy. Suppose the replenishment lead-time is 1 week (7 days), and the weekly demand is normally distributed with mean 250 and standard deviation 64.55. Assume optimal order quantity is 250 units. Part A. Suppose we set R- 282. What is the service level? Part B. What is the average inventory level? Part C. How many stock-out cycles might you expect per year? (Assume 350 days per year) Hint: Again, we ask for stock-out cycles and not stock-out quantities. Think of type 1 service level. Part D. What should the safety stock be if we want to provide a service level of 98%? Part E.Suppose we have two categories of customers, gold and silver. We promise a higher level of service to the gold customers (e.g., 95% of demand is satisfied from stock): for silver customers, the service level would be lower, e.g., 85% of demand satisfied from stock. Describe how you might manage the inventory to serve the two demand classes? (We don't expect you to find a specific policy, but you should attempt to describe conceptually how you might structure a policy and then operate with this policy). Part F. Modify your answer to Part A if we know that the demand is miformly distributed in [138.2, 361.8]; i.e. it has the same mean 250 and standard deviation 64.55.

Explanation / Answer

PLEASE FIND ANSWERS TO FIRST 5 QUESTIONS :

Part A :

Reorder point = Demand during lead time + Safety stock

Or, 282 = 250 + Safety stock

Or, safety stock = 282 – 250 = 32

Also,

Safety stock = Z value x Standard deviation of demand during lead time

Or, 32 = Z x 64.55

Or, Z = 32/64.55 = 0.49 ( rounded to 2 decimal places )

Corresponding value of probability for Z = 0.49 as derived from standard normal distribution table = 0.68793

Therefore service level = Probability x 100 = 68.79 %

SERVICE LEVEL = 68.79 %

PART B:

Average inventory = Optimal order quantity/2 + safety stock = 250/2 + 32 = 125 + 32 = 157 units

AVERAGE INVENTORY = 157 UNITS

PART C :

Stock out probability = 1 – service probability = 1 – 0.68793 = 0.3120

Therefore expected number of stockouts in a year of 350 days = 350 x 0.3120 = 109.2 ( 109 rounded to nearest whole number )

EXPECTED NUMBER OF STOCKOUT CYCLES PER YEAR = 109

PART D :

Z value for service level 98 % i.e. service probability of 0.98 = NORMSINV ( 0.98 ) = 2.053

Thus, Safety stock = Z value x Standard deviation of demand during lead time = 2.053 x 64.55 = 132.52 ( 133 rounded to nearest whole number )

SAFETY STOCK = 133 UNITS

PART E:

For different service levels, inventory will be managed using different levels of safety stock.

Higher the service level required, higher will be requirement for safety stock .

Quantum of safety stock required will be defined as product of z value of service level probability x Standard deviation of demand during lead time .

Thus safety stock requirement for service level of 98 % ( i.e. service probability of 0.98 ) = NORMSINV ( 0.98 ) X standard deviation = 2.053 x 64.55 = 132 ( rounded to nearest whole number )

Safety stock requirement for service level of 95%( i.e. service probability of 0.95) = NORMSINV (0.95) X standard deviation of demand during lead time = 1.6448 x 64.55 = 106.17 ( 106 rounded to nearest whole number )

Thus a safety stock of 132 units have to be maintained for Gold Customers and a safety stock 106 units have to be maintained for Silver customers

SERVICE LEVEL = 68.79 %

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