Exercise 4
A product is stored in a warehouse. Its daily demand is normally distributed with average value equal to 100 units and standard deviation equal to 12 units. The order lead time is equal to 7 days and standard deviation 2 days. The ordering cost is equal to 25$ per order and the yearly unit cost of holding inventory is equal to 0.40 $/unit. The warehouse works 365 days per year. The EOQ model is applied in order to manage the inventory level of the product at issue. Calculate the optimal order quantity and the re‐order point so that there is a probability of 98.78% not to have stockouts during the order lead time.
Table for k calculation:
K

Probability of no stockout

Probability of stockout


k

Probability of no stockout

Probability of stockout

0.00

0.5000

50.00%

2.00

0.9772

2.28%

0.25

0.5987

40.13%

2.25

0.9878

1.22%

0.50

0.6915

30.85%

2.50

0.9938

0.62%

0.75

0.7734

22.66%

2.75

0.9970

0.30%

1.00

0.8413

15.87%

3.00

0.9987

0.13%

1.25

0.8944

10.56%

3.25

0.9994

0.06%

1.50

0.9332

6.68%


3.50

0.9998

0.02%

1.75

0.9599

4.01%


3.75

0.9999

0.01%

Solution:
Given data:
D(daily) = Daily demand = 100 units
_{D } = 12 units
Order lead time = L = 7 days
_{L } = 2 days
Setup or Order cost = S= 25$/order
Operational days= #of period = 365 days
Annual holding cost = H * # of periods = 0.4 $/unit
Probability of not having stockout = P=98.78%
EOQ inventory management model.
By applying EOQ model, we are able to determine the optimal quantity of items to be included in each order. Because, EOQ model follows variable interval for order placing, it requires rigorous (continuous) control and fixed quantity for each order
= = 2136 units/order
Next step is to calculate the reorder point. In order to compute the reorder point we have to apply a formulation that considers the probabilistic demand during lead time and probabilistic lead time.
Reorder point = d * L + Safety stock (SS) = 100 [units/days] * 7 [days] + 456 units = 1156 units
=
The value of k for 98.78 % is given in the table and it is equal to 2.25.
Exercise 5
The weekly demand for a class C product is normally distributed with average value equal to 2,000 units and standard deviation equal to 800 units. The order lead time is also normally distributed with average value equal to 3 weeks and standard deviation equal to 1 week.
The unit selling price is 5 €/unit, the ordering cost is equal to 20 €/order, and the unit cost of holding inventory is equal to 18% of the unit selling price on a yearly basis. Assume 1 year equal to 50 weeks.
What is the best inventory management model? Motivate your answer. In case you choose a fixedtime period model assume the time between inventory reviews equal to 5 weeks.
Calculate the objective inventory level and the safety stock so that the probability of not having stockouts is equal to 95.99%.
Solution:
Given data:
D(weekly) = Weekly demand = 2000 units
_{D }= 800 units
Order lead time = L = 3 weeks
_{L }= 1 week
Setup or Order cost = S= 20 €/order
Operational weeks= # of period = 50 weeks
Unit selling price = 5 €/unit
Annual holding cost = H * # of periods = 18% of Unit selling price = 0.18*5 €/unit = 0.9 €/unit
Probability of not having stockout = P= 96%
Interval time of inventory reviews = T = 5 weeks

Selling price is 5€/unit, which is relative low cost items and classified as Cclass items. For Cclass items it not necessary to have dynamic and strict control of inventory. Therefore, the most suitable inventory management model is Fixedtime model.

For Fixedtime inventory management model, we have to define Objective level of inventory.
Objective level (OL) = d*(L+T) + SS
To define Safety stock the formula will change slightly, we must introduce the time interval between inventory reviews as following:
=
OL = d*(L+T)+SS = 2000*(3+5) +5285 = 21285 units
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