Operations and Supply Chain Management The Core

INVENTORY MANAGEMENT chapter 11 369
where
q = Quantity to be ordered
T = The number of days between reviews
L = Lead time in days (time between placing an order and receiving it)
d ​ ¯ ​ = Forecast average daily demand
z = Number of standard deviations for a specified service probability
σ T + L = Standard deviation of demand over the review and lead time
I = Current inventory level (includes items on-order)
Note: The demand, lead time, review period, and so forth can be any time units such as
days, weeks, or years, as long as they are consistent throughout the equation.
In this model, demand ​(​d ¯ ​ )​can be forecast and revised each review period if desired,
or the yearly average may be used if appropriate. We assume that demand is normally
distributed.
The value of z is dependent on the probability of stocking out and can be found using
Appendix E or by using the Excel NORM.S.INV function.
Example 11.5: Quantity to Order
Daily demand for a product is 10 units with a standard deviation of 3 units. The review period is
30 days, and lead time is 14 days. Management has set a policy of satisfying 98 percent of demand
from items in stock. At the beginning of this review period, there are 150 units in inventory.
How many units should be ordered?
Excel:
Inventory
Control
SOLUTION
The quantity to order is
q = ¯ ​d ​(T + L) + z​σ​
​
T+L ​ − I
​ ​ ​
= 10(30 + 14) + z​σ​ T+L ​ − 150 ​
Before we can complete the solution, we need to find σ T + L and z. To find σ T + L , we use the notion,
as before, that the standard deviation of a sequence of independent random variables equals the
square root of the sum of the variances. Therefore, the standard deviation during the period T + L is
the square root of the sum of the variances for each day:
_____
T+L
​σ​ T + L ​ = ​ √
​ ∑
​ ​σ​ 2 d ​ [11.12]
Because each day is independent and σ d is constant,
the z-score for P = 0.98 is 2.05.
The quantity to order, then, is
________
i = 1
___________
​σ​ T + L ​ = ​√ (T + L)​σ​ 2 d ​ = ​√ (30 + 14) ​(3)​ 2 ​ = 19.90
​q = ¯ ​d ​(T + L) + z​σ​ T + L ​ − I = 10(30 + 14) + 2.05(19.90) − 150 = 331 units
To ensure a 98 percent probability of not stocking out, order 331 units at this review period.
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